Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Find the number of subsets $A$ of $S$ such that $x \in A$ and $2x \in S \implies 2x \in A$. Let $S=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. Find the number of subsets $A\subseteq S$ such that $x\in A$ and $2x\in S$ $\implies 2x\in A$.
My Attempt
I broke the problem into cases. I ... | A subset $A$ of $S$ satisfying your property is exactly the same as a set that satisfies the following properties :
*
*If $A$ contains 1, then $A$ contains $1$,$2$,$4$ and $8$ ;
*If $A$ contains 2, then $A$ contains $2$,$4$ and $8$ ;
*If $A$ contains 3, then $A$ contains $3$, $6$ ;
*If $A$ contains 4, then $A$ con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4584150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Reduction of the Degree of a Curve by a Substitution Let $y^2=P_{2n}(x)$ be an (hyper)elliptic curve, where $P_{2n}$ is a polynomial of degree $2n.$
It is said that the substitution $$x=x_1^{-1}+\alpha\qquad \text{and} \qquad y=y_1x_1^{-n}$$ reduces the degree of the curve to some $y_1^2=P_{2n-1}(x_1).$ Here $\alpha$ i... | TL;DR: For an even model hyperelliptic curve, the line at infinity $Z=0$ intersects the curve in two distinct points. For an odd model, the line at infinity intersects the curve in just one point, and is tangent to the curve at this point. The substitution you've given maps a Weierstrass point $(\alpha, 0)$, at which t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4586828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
For which integer values of $m \geq 0$ and $n \geq 0$ is $A_m(n) = (\frac 2 3) ^ m (n + 2) - 2$ a positive integer? For which integer values of $m \geq 0$ and $n \geq 0$ is $A_m(n) = (\frac 2 3) ^ m (n + 2) - 2$ a positive integer?
I made a table of the first few expressions for $m \in [0, 5]$
$$
\begin{align}
m && A_m... | $\left(\dfrac 2 3\right) ^ m (n + 2) - 2=k\iff 2^m(n+2)=3^m(k+2)$ so if you take
$$n+2=3^m\\k+2=2^m$$ and you get infinitely many solutions..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4587257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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A series sum involving Catalan numbers I was trying to compute $$\sum_{k=0}^{n} \left(-\frac{1}{2}\right)^k \, \binom{2k}{k} \, \frac{k}{k+1} = \sum_{k=0}^n \left(-\frac12\right)^k k C_k$$
(where $C_k$ is the $k^{\rm th}$ Catalan number) but could not come up with a good idea.
I found out it is equal to $$\sum_{k=0}^{... | One can view the Oeis sequences to find A014138 which is a finite sum of Catalan numbers given as $$ a_{n} = \sum_{k=0}^{n} C_{k}. $$ So there is a base case of the series. Now, consider the series
$$ a_{n}(x) = \sum_{k=0}^{n} C_{k} \, x^k $$
which leads to
$$ x \, \frac{d \, a_{n}(x)}{d x} = b_{n}(x) = \sum_{k=0}^{n} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4588459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Minimum square sum of inscripted triangle Problem: In a right triangle $ABC$, the hypotenuse is long $4$ and the angle in $B$ is $30°$. Calling $N$ the midpoint on the side $AB$ (the hypotenuse), and $M$ the middle point on the side $CB$, consider a random point on the side $AC$, lets call it $P$ and $AP = x$. Find the... | I like mindless approaches.
Using Analytical Geometry, you can assign $(x,y)$ coordinates of
*
*C : $~\displaystyle (0,0).$
*B : $~\displaystyle \left(2\sqrt{3},0\right).$
*A : $~\displaystyle \left(0,2\right).$
*M : $~\displaystyle \left(\sqrt{3},0\right).$
*N : $~\displaystyle \left(\sqrt{3},1\right).$
First, c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4591212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Proof that $\binom{n}{k}\frac{1}{n^k} \le \frac{1}{k!}$ In a Calculus textbook, I was faced with the following
Problem: If $n$ is a natural number with $n \ge 1$, proof that $\binom{n}{k}\frac{1}{n^k} \le \frac{1}{k!}$ for all $k \in \mathbb{N}$.
The book presented a different solution than the one, I came up with. So ... | Here is an alternative method:
We want to show $\frac{n!}{(n-k)!}\le n^k$. The LHS is the number of ordered subsets of size $k$ of a set of $n$ elements, the RHS is the number of ordered lists of size $k$ taken from a set of $n$ elements, where its ok if two elements are equal. Clearly the inequality is satisfied with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4596198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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To prove $1^1\cdot2^2\cdot 3^3...\cdot n^n<(\frac{2n+1}{3})^{\frac{n(n+1)}{2}} $ So we have to prove the following for $n\in N $ $$1^1\cdot 2^2\cdot 3^3...\cdot n^n<\left(\frac{2n+1}{3}\right)^{\frac{n(n+1)}{2}} $$
So I used concept of weighted means (arithmetic and geometric) used AM GM inequality.
$$AM=\frac{a_1w_1+a... | This is not a proof since working for large values of $n$
$$\prod_{k=1}^n k^k=H(n)$$ where $H(n)$ is the hyperfactorial function.
Expanding its logarithm
$$\log (H(n))=-\frac 14 n^2+\frac 1{12} \left(6 n^2+6 n+1\right)\log(n)+\log (A)+\frac{1}{720 n^2}\left(1-\frac{1}{7 n^2}+\frac{1}{14
n^4}+O\left(\frac{1}{n^6}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4598722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Rational approximation of $\pi$ by recursion I wonder if the following result is already known and may be considered as interesting : let $\mathcal{C}$ be the real algebra of continuous functions $f:\left[0,1\right]\to\mathbf{R}$, $T:\mathcal{C}\to \mathcal{C}$ the map defined by
$$
T\left(f\right)\left(x\right)=f\left... | Too long for a comment.
Using the Gaussian hypergeometric function
$$I_n = \frac{1}{2} \sum_{k=0}^{2^n-1} \frac{2^k}{\left(2k+1\right)\binom{2k}{k}}=\frac \pi 4-J_n$$
$$J_n=\frac{2^{2^n-1}}{\left(2^{n+1}+1\right) \binom{2^{n+1}}{2^n}}\,\,\, _2F_1\left(1,2^n+1;2^n+\frac{3}{2};\frac{1}{2}\right)$$
$$J_n < \frac{2^{2^n}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4602604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find a line that passes through a point A and cuts two other lines. Let $L_1$ be the straight line in $R^3$ given by (x, y, z) = (2, 2, 0) + t(3, 0, 2). A plane containing the line $L_1$ and the point A = (8, 2, 3)is given by (x,y,z) = (2,2,0) + s(6,0,3) + t(3,0,2)
The line $L_2$ is given by (x, y, z) = (5, 1, 0) + $\t... | $L_1: (2, 2, 0) + t (3,0,2) $
$L_2: (5,1,0) + s(2,1,1) $
$ A = (8,2,3) $
The vector from $A$ to a point on $L_1$ must be a multiple of the vector from $A$ to a point on $L_2$. Thus define the first vector as follows
$ V_1 = (2, 2, 0) + t (3, 0, 2) - (8, 2, 3) = (-6, 0, -3) + t (3, 0, 2) $
and the second vector
$ V_2 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4603002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Why is $2\sqrt{x+1}-2\geq\frac{\sqrt{x}}{2}$ for $x \in \mathbb{N}$? I have found following formula stands:
$2\sqrt{x+1}-2\geq\frac{\sqrt{x}}{2}$
($x \in \mathbb{N}$)
I have wondered why this is true. I have checked its rightness on the graph drawing tools, but I want to know the mathematical proof of this.
Although I ... | Using the fact that $\sqrt{a}-\sqrt{b}=\dfrac{a-b}{\sqrt{a}+\sqrt{b}}$ , we have
$$\begin{aligned}2\sqrt{x+1}-\frac{\sqrt{x}}{2}&=\frac12\left(4\sqrt{x+1}-\sqrt{x}\right)\\
&=\frac12\cdot\frac{16(x+1)-x}{4\sqrt{x+1}+\sqrt{x}}\\
&=\frac{15x+16}{2(4\sqrt{x+1}+\sqrt{x})}\\
&\geqslant\dfrac{15(x+1)}{10\sqrt{x+1}}\\
&=\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4604944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Assuming x is small, expand $\frac{\sqrt{1-x}}{\sqrt{1+2x}}$ up to and including the term in $x^{2}$ I have tried this many times but can't quite land on the correct answer.
The correct answer:
$1-\frac{3x}{2}+\frac{15x^{2}}{8}$
These are the steps I took:
*
*Re wrote it as: $\left ( 1-x \right )^{\frac{1}{2}}\left (... | $\frac{1}{1+2x}=1-2x+4x^2-8x^3+O(x^4)$
so
$\frac{1-x}{1+2x}=1-2x+4x^2-8x^3-x+2x^2-4x^3+8x^4+O(x^5)=$
$1-3x+6x^2-12x^3+8x^4+O(x^5).$
Therefore, when $x$ is small,
$\sqrt{\frac{1-x}{1+2x}}\sim 1+\frac{1}{2}\left(-3x+6x^2-12x^3+8x^4\right)-\frac{1}{8}\left(-3x+6x^2-12x^3+8x^4\right)^2\sim$
$1-\frac{3}{2}x+3x^2-\frac{9}{8}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4605578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Evaluating a double integral I was trying to evaluate the following integral
$$\int_{x=0}^{\infty}\int_{y=0}^{\infty}\frac{y \ln y \ln x}{(x^2+ y^2)( 1+y^2)} dy dx.$$
I have a guess that the value of this integral is $\frac{\pi^4}{8}$. But I am unable to prove it.
Could someone please help me in evaluating this integra... | To show your identity holds, make the one-dimensional change of variables $x=ty$ to get
\begin{align*}
\int_{0}^\infty \int_0^\infty \frac{y \ln(y) \ln (x)}{(y^2+1)(x^2+y^2)} \ dy \ dx &= \int_{0}^\infty \int_0^\infty \frac{y^2 \ln(y) \ln (ty)}{(y^2+1)(t^2+1) y^2} \ dy \ dt \\
&=\int_{0}^\infty \int_0^\infty \frac{ \ln... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4606092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
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A line $4x+y-1=0$ through $A(2,-7)$ meets the line $BC$ whose equation is $3x-4y-1=0$ at point $B$. Find Equation of Line $AC$, such that $AB=AC$. A line $4x+y-1=0$ through $A(2,-7)$ meets the line $BC$ whose equation is $3x-4y-1=0$ at point $B$. Find Equation of Line $AC$, such that $AB=AC$.
My Approach:
I took slope ... | Solve the system of equations corresponding to the two given lines to find the coordinates of their point of intersection, which is $B.$ Find the distance between $A$ and $B.$ We'll call this distance $r.$ Solve the system of equations $$3x-4y=1,(x-2)^2+(y+7)^2=r^2$$. There are two solutions. One solution is the point ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4606263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How do we prove $x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0$? Question
How do we prove the following for all $x \in \mathbb{R}$ :
$$x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0 $$
My Progress
We can factorise the left hand side of the desired inequality as follows:
$$x^6+x^5+4x^4-12x^3+4x^2+x+1=(x-1)^2(x^4+3x^3+9x^2+3x+1)$$
However, after... | Your quartic polynomial is called self-reciprocal or palindromic, in mathematics.
Let,
$$
\begin{align}P(x):=&x^4+3x^3+9x^2+3x+1\end{align}
$$
The case $x=0$ is trivial. Therefore, we can divide all terms of the polynomial $P(x)$ by $x^2\thinspace : (x\neq 0\thinspace)$
$$
\begin{align}\frac {P(x)}{x^2}=&\left(x^2+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4611737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 3
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How do I integrate this rational function? Evaluate this integral
$$I=∫\dfrac{1-x^2}{x^4+x^2+1}dx$$ (with $x$ is different from 0).
I tried divided both the numerator and denominator by $x^2$, and I got $$∫\dfrac{\frac{1}{x^2}-1}{x^2+\frac{1}{x^2}+1}dx.$$
I then put $t=x+1/x$ and got this $$I=∫-\dfrac{dt}{t^2-1}.$$
Fin... | Since xpaul answered your question. Notice that another way is using partial fraction and we get
$$\frac{1-x^2}{x^4+x^2+1}=\frac{1}2\underbrace{\left(\frac{1+2x}{1+x+x^2}\right)}_{(*)}+\frac{1}{2}\underbrace{\left(\frac{1-2x}{1-x+x^2}\right)}_{(**)}$$
For $(*)$ substitution $u(x)=1+x+x^2$ and for $(**)$ substitution $v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4612912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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If $α,β,γ$ are the roots of the equation $f(x)=x^3+qx+r=0$ then find the equation whose roots are, If $α,β,γ$ are the roots of the equation $f(x)=x^3+qx+r=0$ then find the equation whose roots are, $\frac{\beta^2+\gamma^2}{\alpha^2}$,$\frac{\alpha^2+\gamma^2}{\beta^2}$,$\frac{\beta^2+\alpha^2}{\gamma^2}$.
My solution g... | To answer your question, rather than suggest an alternative method: the error is in "Thus, $\frac{2r}{a-1}+\frac{r(a+1)}{(1-a)}+r=0$, which implies $a^2-a-2=0$". In fact (since $a \ne 1$) it implies that $2r -r(a+1) +r(a-1) = 0$, which is identically true. Instead, use your (correct) expression for $\gamma^3$ and subst... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4613589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Tricky integral in a complex plane How to integrate using residues:
$$\oint\limits_{|z|=2}\frac{1}{(z^6-1)(z-3)} dz $$
if the idea probably requires to change the sum of 6 residues to aditive inverse of residue at infinity.
I believe that the sum of 6 residues is:
$$2 \pi i\sum_{i=1}^6 Res_i \frac{1}{(z^6-1)(z-3)}=-2\p... | Thank you for adding your work. Expanding on my comment: for $|z|\ge R\gt3$, we have
$$
\begin{align}
\left|\frac1{\left(z^6-1\right)(z-3)}\right|
&\le\frac1{|z|^7}\frac1{\left(1-\frac1{|z|^6}\right)\left(1-\frac3{|z|}\right)}\tag{1a}\\
&\le\frac{\frac{R^6}{R^6-1}\frac{R}{R-3}}{|z|^7}\tag{1b}
\end{align}
$$
For $R=7$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4614189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Find the maximum value of $xy+yz+xz-2xyz$
If $x+y+z=1$ and $0\le x,y,z\le1$ then find the maximum value of expression $$xy+yz+xz-2xyz$$
Solution that I have
$$(1-2x)(1-2y)(1-2z)=1-2\sum x+4\sum xy-8xyz=4\sum xy-8xyz-1$$
$\implies$
$$\sum xy-2xyz=\frac{1+(1-2x)(1-2y)(1-2z)}{4}$$
Now
$$\frac{\sum(1-2x)}{3}\ge\{(1-2x)(1... | Another way for proving that $f(x,y,z)=xy+yz+zx-2xyz\leq \frac{7}{27}$ if $x,y,z\in(0,1)$ and $x+y+z=1.$ We denote $x=X+1/3,y=Y+1/3, z=Z+1/3$ with $X,Y,Z\in [-1/3.2/3]$ and $X+Y+Z=0.$ We denote
$$A=-2(XY+YZ+ZX)=X^2+Y^2+Z^2$$ leading to
$$f(x,y,z)=\frac{7}{27}-\frac{1}{6}(A+12XYZ).$$ If 2 of the three numbers $X, Y,Z$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4615444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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Finding the value of given integral. It was asked to find the correct option(s) for the given integral: $$I_n = \displaystyle\int_{\frac{n}{2}}^{\frac{n+1}{2}}\dfrac{\sin{(\pi(\sin^2{\pi x}}))}{(\sqrt2)^x} \, dx$$
(a)$\dfrac{I_n}{I_{n+4}}=2$
(b)$\dfrac{I_n}{I_{n+4}}=\dfrac{1}{\sqrt2}$
(c)$\dfrac{\displaystyle\sum_{n=0}... | First, notice that for any even integer $2k$
$$I_{n+2k} = \int_{\frac{n}{2}+k}^{\frac{n+1}{2}+k}\frac{\sin\left[\pi \sin^2(\pi x)\right]}{\left(\sqrt{2}\right)^x}dx = \int_{\frac{n}{2}}^{\frac{n+1}{2}}\frac{\sin\left[\pi \sin^2(\pi x + \pi k)\right]}{\left(\sqrt{2}\right)^{x+k}}dx = \frac{1}{\left(\sqrt{2}\right)^k}I_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4615702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
$A=\begin{pmatrix}a&c&-1\\ 1-c&-a&0\\ 5&3&b\end{pmatrix}$ has an eigenvector $(-1,1,-1)'$, and $|A|=0$, can we find $a,b,c$?
$A=\begin{pmatrix}a&c&-1\\ 1-c&-a&0\\ 5&3&b\end{pmatrix}$ has an eigenvector $(-1,1,-1)'$, and $|A|=0$, can we find $a,b,c$?
By $Ax=tx$ for some $t$, we see $c=1+t-a, b=t-2$ (with the other equ... | Since $\det(A)=0$, then we have
$$-ba^{2}-5a+(c-1)(bc+3)=0,\quad (1)$$
Since $(-1,1-1)$ is an eigenvector for some eigenvalue $k$ of $A$ we have
$$-a+c+k=-1,\quad (2)$$
$$-a+c-k=1,\quad (3)$$
$$-b+k=2,\quad (4)$$
Solving the non-linear system with equations $(1),(2),(3)$ and $(4)$ with Mathematica we have one solution
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4616704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show that $|\tan x-x|\leq 8x^2$ if $|x|\leq \pi/3$ Show that $|\tan x-x|\leq 8x^2$ if $|x|\leq \pi/3$. I think this is supposed be solved using the maclaurin series.
Let $f(x)=\tan x$ then $f'=1/\cos ^2x$ and $f''=\frac{-2\cos x \sin x}{\cos^4 x}=\frac{-\sin 2x}{\cos^4 x}$. Since $f(0)=0$ and $f'(0)=1$ we have that
$|x... | Let
$$ f(x)=\tan x-x. $$
Then there is $\theta\in(0,1)$ such that
$$ f(x)-f(0)=f'(\theta x)x. $$
Note
$$ f'(\theta x)=\sec^2(\theta x)-1=\tan^2(\theta x)=\frac{\sin^2(\theta x)}{1-\sin^2(\theta x)}. $$
Using
$$ \frac{2}{\pi}\theta\le\sin\theta\le\theta, \theta\in[0,\frac\pi2]$$
one has
$$ \sin^2(\theta x)\le \theta^2x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4616878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to order $x = \sqrt{3}-1, y = \sqrt{5}-\sqrt{2}, z = 1+\sqrt{2}$ ascendingly? How would I order $x = \sqrt{3}-1, y = \sqrt{5}-\sqrt{2}, z = 1+\sqrt{2} \ $ without approximating the irrational numbers? In fact, I would be interested in knowing a general way to solve such questions if there is one.
What I tried to so... | Suppose $ x \geqslant y $, then
$$ \begin{array} { r c l }
\sqrt3 + \sqrt2 &\geqslant& \sqrt5 + 1 \\
(\sqrt3 + \sqrt2)^2 &\geqslant& (\sqrt5 + 1)^2 \\
5 + 2 \sqrt6 &\geqslant& 6 + 2 \sqrt5 \\
2( \sqrt6 - \sqrt5) &\geqslant& 1 \\
2( \sqrt6 - \sqrt5)( \sqrt6 + \sqrt5) &\geqslant& ( \sqrt6 + \sqrt5) \\
2(6 - 5) &\geqslant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4617031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Prove that $\lfloor {(\frac{\sqrt{5}+1}{2})}^{4n-2}\rfloor-1$ is a square number where $n$ is a natural number. I found this problem in a junior high school math competition.
Prove that $\lfloor {(\frac{\sqrt{5}+1}{2})}^{4n-2}\rfloor-1$ is a
square number where $n$ is a natural number.
Here's what I think
Suppose $x$... | Not to be confused with this exceedingly similar question.
I'll reproduce the solution found here (with a few added details).
Because $\frac{\sqrt{5} - 1}{2} < 1$, we also know that $\left(\frac{\sqrt{5} - 1}{2}\right)^{4n - 2} < 1$. Then
$$\begin{align}
\left\lfloor\left(\frac{\sqrt{5} + 1}{2}\right)^{4n - 2}\right\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Derivative of $(2x^2+3)^2(x^3-2x)^4$ I want to know if there's another method in shorter way. I came up to this problem
Find the $f'(x)$
$$
f(x) = (2x^2+3)^2\cdot(x^3-2x)^4
$$
Applying product rules
$$
\frac{d}{dx}f(x)g(x) = f(x)\frac{d}{dx}g'(x) + g'(x)\frac{d}{dx}f(x)
$$
Solve:
$$ f = (2x^2+3)^2 $$
$$ g = (x^3-2x)^... | Your method is correct, and since the question states that you should make use of the product rule, it is the right solution.
If you're new to derivatives, you may not yet have learned about derivatives of logs or about implicit differentiation; however you're clearly already familiar with the chain rule, so you should... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
BMO2 1994 Q1 - Need Help Find the first number such that the average of the sum of the squares from $1$ to $n$ (where $n > 1$) equals $k^2$.
Here is what I have done so far:
The sentence is equivalent to saying that $\frac{\sum_{r=1}^{n}r^2}{n} = k^2$. Knowing the sum of the squares formula, the average of the sum of t... | Even tough the other answer provided is far more elegant, I would like to show another possible approach:
First of all, we need $\frac{(n+1)(2n+1)}{6}$ to be an integer, and checking the possibilities modulo $2$ and $3$ we get that $n$ must be of the form $6a\pm 1$. However, if $n$ were of the form $6a-1$, plugging it ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4621711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluation of $\lim_{n\to\infty}\sum_{k=1}^{n}kA^{k-1}$ Let
$$\begin{align}
A:&=\begin{pmatrix}{11\over 6}&2\\-1&-1\end{pmatrix}\\
B:&=\lim_{n\to\infty}\sum_{k=1}^{n}kA^{k-1}\\
A^{0}&=I
\end{align}$$
I want to evaluate $B$.
My tries:
$$\begin{align}
A&=\begin{pmatrix}{11\over 6}&2\\-1&-1\end{pmatrix}\\
&={6\over 6}\beg... | We have
$$
(I-2A+A^2)\sum_{k=1}^{n} kA^{k-1}
= \sum_{k=1}^{n} kA^{k-1}
-2\sum_{k=1}^{n} kA^k
+\sum_{k=1}^{n} kA^{k+1} \\
=I+2A+\sum_{k=3}^{n} kA^{k-1} \\
-2\left(A+\sum_{k=2}^{n-1} kA^k+nA^n\right) \\
+\sum_{k=1}^{n-2} kA^{k+1}+(n-1)A^n+nA^{n+1} \\
=I+2A-2A-2nA^n+(n-1)A^n+nA^{n+1} \\
+\sum_{k=2}^{n-1} (k+1)A^k
-2\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4622717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Determine whether this series converges: Determine whether this series converges:
$ \sum_{n=1}^{\infty}\frac{\sqrt n\cos(n^2)}{n^{4/3}+\cos(n^2) }$
I tried
$$-1\leq \cos(n^2) \leq 1 \Rightarrow \sum_{n=1}^{\infty}\frac{- \sqrt n}{n^{4/3}-1 } \leq \sum_{n=1}^{\infty}\frac{\cos(n^2)\cdot \sqrt n}{n^{4/3}+\cos(n^2) } \l... | Denote
\begin{align}
a_n = \frac{\sqrt{n}}{n^{4/3} +\cos(n^2)}, \quad b_n = \cos(n^2), \quad \text{ and } \quad B_N = \sum^N_{n=0} \cos(n^2).
\end{align}
Then the given series can be expressed as
\begin{align}
\sum^\infty_{n=0} a_nb_n.
\end{align}
Now, using Abel's summation formula (or the so called summation-by-part... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Find the coefficient of $f(x)=e^x \sin x$ for the 5th derivative $f^{(5)}(0)$ using maclaurin series
Find the coefficient of $f(x)=e^x \sin x$ for the 5th derivative $f^{(5)}(0)$ using maclaurin series
the maclaurin polynomial is supposed to be $M_5(0)=f(0)+ \frac{f'(0)}{1!}+...+ \frac{f^{(5)}(0)}{5!}$
We know that ... | You have that
$$e^x=1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\frac{1}{4!}x^4+\text{higher order terms}$$
and
$$\sin x=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5+\text{higher order terms}.$$
If we multiply these together and just look at the resulting $x^5$-term, we get
$$e^x\sin x=\text{lower order terms}+\left(\frac{1}{4!}\cdot1-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4627843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Range of Trigonometric function having square root
Finding range of function
$\displaystyle f(x)=\cos(x)\sin(x)+\cos(x)\sqrt{\sin^2(x)+\sin^2(\alpha)}$
I have use Algebric inequality
$\displaystyle -(a^2+b^2)\leq 2ab\leq (a^2+b^2)$
$\displaystyle (\cos^2(x)+\sin^2(x))\leq 2\cos(x)\sin(x)\leq \cos^2(x)+\sin^2(x)\cdo... | Let $a=|\sin\alpha|.$
First notice that (since $|\sin x|\le\sqrt{a^2+\sin^2x}$) when the maximum value $f_{\max}$ is attained the $\sin$ and $\cos$ of $x$ are positive, and that the minimum value is $f_{\min}=-f_{\max},$ corresponding to the same $\sin$ and the opposite $\cos.$
$$f'(x)=\left(1+\frac{\sin x}{\sqrt{a^2+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Is it true that if $a, b, x, y>0$, $a+b \ge x+y$ and $ab \le xy$ then $a^n+b^n≥x^n+y^n$ for $n>1$? Let $a, b, x, y>0$, $a+b \ge x+y$ and $ab \le xy$ then $a^n+b^n \ge x^n+y^n$ where $n=2$.
Is it true that if $a, b, x, y>0$, $a+b \ge x+y$ and $ab \le xy$ then $a^n+b^n \ge x^n+y^n$ for $n \ge 1$?
I have just been che... | Let
$d_n
= a^n+b^n
$
and let $a+b=d$.
$\begin{array}\\
d_n(a+b)
&=(a^n+b^n)(a+b)\\
&=a^{n+1}+ba^n+ab^n+b^{n+1}\\
&=a^{n+1}+b^{n+1}+ab(a^{n-1}+b^{n-1})\\
&=d_{n+1}+abd_{n-1}\\
\text{so}\\
d_{n+1}
&=d_n(a+b)-abd_{n-1}\\
&=dd_n-abd_{n-1}\\
\end{array}
$
Similarly,
if
$e_n=x^n+y^n
$
and
$e=x+y$,
$e_{n+1}
=ee_n-xye_{n-1}
$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Quadratic polynomials $f(x)$ with rational coefficients such that $f(p)=q, f(q)=r, f(r)=p$ where $p,q,r$ are the roots of $x^3-7x+7$. Let $p,q,r$ be the three roots of $x^3-7x+7$. Find all quadratic polynomials $f(x)$ with rational coefficients such that $f(p)=q, f(q)=r, f(r)=p$.
So far, I let $f(x)=ax^2+bx+c$, and I ... | Multiply each of your three equations by the root on its left hand side
$$ap^3+bp^2+cp=pq,$$
$$aq^3+bq^2+cq=qr,$$
$$ar^3+br^2+cr=rp.$$
Add the three equations again and apply Vieta's formulas, but also using $p^3 = 7p-7$ etc (they are roots!), so $$p^3+q^3+r^3 =-21$$ gives $-21a+14b=-7$, so $3a=2b+1$, which now gives a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4632986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Bounds on $\max_i p_i$ in terms of $\sum_i p_i^2$ Suppose $(p_1,p_2,\ldots,p_d)$ is a discrete probability distribution over $d$ outcomes with $p_i\ge p_{i+1}$
Given the value of $\rho=\sum_{i=1}^d p_i^2$ (aka purity), what are the possible values of $p_1$?
I'm particularly interested in "large-d" regime.
Below is the ... | The range of $p_1$ is given by $[\alpha, \beta]$ where
$$\beta = \frac{1}{d} + \sqrt{\rho - \frac{\rho}{d} - \frac{1}{d} + \frac{1}{d^2}}$$
and
$$\alpha = \frac{1}{m} + \frac{1}{m}\sqrt{\frac{m\rho - 1}{m-1}}$$
where $m = \lfloor 1/\rho\rfloor + 1$.
Moreover, $p_1 = \beta$ if $p_2 = p_3 = \cdots = p_d = \frac{1}{d} - \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Determining whether $\frac{1}{2^{2}} + \frac{2}{3{^2}} + ... +\frac{n}{(n+1)^{2}}$ is a Cauchy sequence. My attempt so far: If $n \leq m$, then
$a_{m} - a_{n} = \frac{n+1}{(n+2)^2}+\frac{n+2}{(n+3)^2}+...+\frac{m}{(m+1)^2} < \frac{n+1}{(n+2)(n+1)}+\frac{n+2}{(n+3)(n+2)}+...+\frac{m}{(m+1)m} = \frac{1}{(n+2)}+\frac{1}{(... | Try working with inequalities in the other directions:
$$a_{m} - a_{n} =\sum_{k=n+1}^m \frac{k}{(k+1)^2}>\sum_{k=n+1}^m \frac{k}{(k+k)^2}=\frac{1}{4}\sum_{k=n+1}^m \frac{1}{k}>\frac{1}{4}\sum_{k=1}^m \frac{1}{k}-\frac{n}{4}$$
Now what do you know about the sum $\sum_{k=1}^\infty \frac{1}{k}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4635140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Inequality $\frac{a^3}{3ab^2+2c^3} +\frac{b^3}{3bc^2+2a^3} +\frac{c^3}{3ca^2+2b^3} \geq \frac{3}{5} $ I have trouble with solving this inequality:
Prove $\frac{a^3}{3ab^2+2c^3} +\frac{b^3}{3bc^2+2a^3} +\frac{c^3}{3ca^2+2b^3} \geq \frac{3}{5}$ for a,b,c>0.
Using Cauchy-Schwartz I got this: $\frac{a^3}{3b^3+c^3} +\frac{b... | By the Cauchy-Schwartz inequality, we have:
$$(\frac{a^3}{3ab^2+2c^3}+\frac{b^3}{3bc^2+2a^3}+\frac{c^3}{3ca^2+2b^3})(a(3ab^2+2c^3)+b(3bc^2+2a^3)+c(3ca^2+2b^3)) \\ \geq (a^2+b^2+c^2)^2 \\ \implies \frac{a^3}{3ab^2+2c^3}+\frac{b^3}{3bc^2+2a^3}+\frac{c^3}{3ca^2+2b^3} \geq \frac{(a^2+b^2+c^2)^2}{3(a^2b^2+a^2c^2+b^2c^2)+2(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4636361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Prove that there is no triple $(a,b,c)$ Prove that there is no triple $(a,b,c)$ such that
$$a,b,c \geq 0,ab+bc+ca=2+abc \textrm{ and } abc \in (1;10+6\sqrt{3}).$$
I transform the expression as follows
$$ab = \frac{m}{c},a + b = \frac{{2 + m - \frac{m}{c}}}{c}.$$
I expect the following inequality to be false and thus co... | $a,b,c \geq 0, \; ab+bc+ca=2+abc, \; abc \in (1,10+6\sqrt{3})$
By AM-GM inequality, $\frac{ab+bc+ca}{3} \geq (abc)^{\frac{2}{3}} \implies ab+bc+ca \geq 3(abc)^{\frac{2}{3}} \implies 2+abc \geq 3(abc)^{\frac{2}{3}}$
Consider $abc = x^3$, then $x^3-3x^2+2 \geq 0 \implies (x-1)(x-1+\sqrt{3})(x-1-\sqrt{3}) \geq 0$
It can b... | {
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"source": "stackexchange",
"question_score": "1",
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Hints on solving $x^{2x}-(x^2+x)x^x+x^3=0$
Solve this equation over $\mathbb{R}^+$: $x^{2x}-(x^2+x)x^x+x^3=0$
I’ve been trying to solve this exponential equation but can’t get the answer because normal substitution ($y=x^x$) isn’t working. Any tips/hints that don’t use logs?
(the section of the book I got this from i... | There are only three obvious solutions
$x=-1,x=1$ and $x=2$
The other intervals are inconsistent with the sign of the simplified equation:
$x^{2x}-(x^2+x)x^x+x^3$ = 0 { expand }
$x^3+x^{2x}-x^{1+x}-x^{2+x}=0$ {a}
$1+x^{-3+2x}-x^{-2+x}-x^{-1+x}=0$ Divide {a} by $x^3$
Simplified to
$-x^{x-2}-x^{x-1}+x^{x^2-3}=-1$ (... | {
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"url": "https://math.stackexchange.com/questions/4641931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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I need to solve $yy'' +y^2 = C$ (constant) Is there an analytic solution? - if so please show me how to derive it.
If not, then boundary conditions for a numerical solution would be $y(0) = 35$, $y(400) = 8$.
I am pretty new at this . . . so apologies if further information/conditions/explanation required. I have acces... | I don't think there is a closed form solution that is not a constant, but you can still find a solution.
It is an ODE of the form $f\left( y\left( x \right),\, \frac{\operatorname{d}y\left( x \right)}{\operatorname{d}x},\, \frac{\operatorname{d}^{2}y\left( x \right)}{\operatorname{d}x^{2}} \right) = 0$ where you can re... | {
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"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Proof that $n^3+2n$ is divisible by $3$ I'm trying to freshen up for school in another month, and I'm struggling with the simplest of proofs!
Problem:
For any natural number $n , n^3 + 2n$ is divisible by $3.$
This makes sense
Proof:
Basis Step: If $n = 0,$ then $n^3 + 2n = 0^3 +$
$2 \times 0 = 0.$ So it is divisi... | RTP : $a^3+3a$ is in the form of $3k$
Proof : $a^3+2a=
a^3-a+3a=
a(a^2-1)+3a=
a(a+1)(a-1)+3a$
We know that sum of three consecutive numbers is divisible by 3 $$3k+3a=3(k+a)=3k$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 13,
"answer_id": 12
} |
Modular exponentiation using Euler’s theorem How can I calculate $27^{41}\ \mathrm{mod}\ 77$ as simple as possible?
I already know that $27^{60}\ \mathrm{mod}\ 77 = 1$ because of Euler’s theorem:
$$ a^{\phi(n)}\ \mathrm{mod}\ n = 1 $$
and
$$ \phi(77) = \phi(7 \cdot 11) = (7-1) \cdot (11-1) = 60 $$
I also know from usin... | You can use exponentiation by squaring (all operations are modulo 77):
$27^{41} = 27^{32+8+1} = 27 \cdot 27^8 \cdot (27^8)^4 = (*)$
$\big[ 27^8 = ((27^2)^2)^2 = (36^2)^2 = 64^2 = 15 \big]$
$(*) = 27 \cdot 15 \cdot 15^4 = 27 \cdot 15 \cdot (15^2)^2 = 27 \cdot 15 \cdot 71^2 = 27 \cdot 15 \cdot 36 = 27$
This uses only 7 m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
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Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
| If you knew of the geometric series, you would know that
$$\frac{1-r^{n+1}}{1-r}=1+r+r^2+r^3+\dots+r^n$$
If we differentiate both sides, we have
$$\frac{nr^{n+2}-(n+1)r^{n+1}+r}{(1-r)^2}=1+2r+3r^2+\dots+nr^{n-1}$$
Letting $r\to1$ and applying L'Hospital's rule on the fraction, we end up with
$$\frac{n(n+1)}2=1+2+3+\dot... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "136",
"answer_count": 36,
"answer_id": 1
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Help in getting the Quadratic Equation I'm starting a chapter on Functions and they had the steps shown to reach the p-q equation.
$$ x_{1,2} = -\frac{p}{2} \pm\sqrt{\left(\frac{p}{2}\right)^2 - q}$$
So I wanted to do the same with the Quadratic Equation. I'm using the base linear equation
$$ax+by+c = 0.$$
The solutio... | Sorry I don't know how to do tex on websites, but I'm trying to learn.
You just made a small mistake on the final step. In the second to last step, we actually have our full equation as:
$$\frac{-b}{2a}\pm \sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}$$
Now we can collect the common facto... | {
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"url": "https://math.stackexchange.com/questions/2796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding the $N$-th derivative of $f(x)=\frac {x} {x^2-1}$ I'm practicing some problems from past exams and found this one:
Find the n-th derivative of this function:
$$f(x)=\frac {x} {x^2-1}$$
I have no idea how to start solving this problems. Is there any theorem for finding nth derivative?
| related problem: (I). I am referring you to this book where you will find the complete answer to the problem of finding the nth derivative of rational polynomials and other classes of functions.
Added Some people suggested the post should be self-contained. Here is the example I am referring to
\begin{equation}
f(x) = ... | {
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"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 5
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Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$ Find all polynomials $p(x)$ such that for all $x$, we have $$(x-16)p(2x)=16(x-1)p(x)$$
I tried working out with replacing $x$ by $\frac{x}{2},\frac{x}{4},\cdots$, to have $p(2x) \to p(0)$ but then factor terms seems to create a problem.
Link: http://web.mit.edu/rwbar... | Putting $x=1$ and $x=16$ we see that $p(2)=p(16) = 0$.
Let $p(x) = (x-2)(x-16)g(x)$.
Thus we see that
$$(x-16)(2x-2)(2x-16)g(2x) = 16(x-1)(x-2)(x-16)g(x)$$
Thus $$ (x-8)g(2x) = 4(x-2)g(x)$$
We now see that $g(4) = g(8) = 0$
Thus $g(x) = (x-4)(x-8)h(x)$
Thus
$$ (x-8)(2x-4)(2x-8)h(2x) = 4(x-2)(x-4)(x-8)h(x)$$
i.e
$$ h(2x... | {
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"url": "https://math.stackexchange.com/questions/3888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 3
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Finding a Function That Approaches Another Function One of my math professors gave me the following challenge. It isn't graded, it's just for fun.
Consider the function:
\begin{equation*}
f_n(x)=x+3^3x^3+5^3x^5+...+(2n-1)^3x^{2n-1},~x \in (0, 1).
\end{equation*}
I want to find which of the following functions $f_n$ is ... | Here's yet another way to verify that it's (b). Let Mathematica do partial fractions decomposition:
Apart[x(x^2+1)(x^4+22x^2+1)/(x^2-1)^4]
This gives $\frac{1}{2 (x+1)}-\frac{7}{2(x+1)^2}+\frac{6}{(x+1)^3}-\frac{3}{(x+1)^4}+\frac{1}{2 (x-1)}+\frac{7}{2(x-1)^2}+\frac{6}{(x-1)^3}+\frac{3}{(x-1)^4}$, which can be expande... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$? I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
| By my post, $$
\cos \frac{\pi}{5}-\cos \frac{2 \pi}{5}=\frac{1}{2}
$$
By $\cos (\pi-\dfrac{\pi}{5} )=-\cos \dfrac{\pi}{5}, $ we have
$$
-\cos \frac{4 \pi}{5}-\cos \frac{2 \pi}{5}=\frac{1}{2}
$$
By double-angle formula, $$
-\left(2 \cos ^{2} \frac{2 \pi}{5}-1\right)-\cos \frac{2 \pi}{5}=\frac{1}{2} \Leftrightarrow
4 \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/7695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 11,
"answer_id": 10
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Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ Using $\text{n}^{\text{th}}$ root of unity
$$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$
Prove that
$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
| $$ \begin{align*}
P & = \prod_{k=1}^{n-1}\sin(k\pi/n) \\
& = (2i)^{1-n}\prod_{k=1}^{n-1}(e^{ik\pi/n}-e^{-ik\pi/n}) \\
& = (2i)^{1-n} e^{-i \frac{n(n-1)}{2}\frac{\pi}{n}} \prod_{k=1}^{n-1}(e^{2ik\pi/n}-1) \\
& = (-2)^{1-n}\prod_{k=1}^{n-1}(\xi^k-1) \\
& = 2^{1-n}\prod_{k=1}^{n-1}(1-\xi^k) \\
\end{align*}
$$
where $\xi=e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "76",
"answer_count": 3,
"answer_id": 0
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How to solve this equation for $x$? How do I solve for $x$ from this equation?
$$ -\frac{1}{x^2} + \frac{9}{(4-x-y)^2} = 0.$$
I need to get this into $x=$"blah"?
| If you add $\frac{1}{x^{2}}$ to both sides of your equation
$$-\frac{1}{x^{2}}+\frac{9}{\left( 4-x-y\right) ^{2}}=0\qquad (1)$$
you get this equivalent one (provided that $\frac{1}{x^{2}}$ is finite,
i.e $x\neq 0$)
$$\frac{1}{x^{2}}=\frac{9}{\left( 4-x-y\right) ^{2}}.\qquad (2)$$
It is satisfied if the square root of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Expressions of $\sin \frac{A}{2}$ and $\cos \frac{A}{2}$ in terms of $\sin A$ I am trying to understand the interpretation of $\sin \frac{A}{2}$ and $\cos \frac{A}{2}$ in terms of $\sin A$ from my book, here is how it is given :
We have $ \bigl( \sin \frac{A}{2} + \cos \frac{A}{2} \bigr)^{2} = 1 + \sin A $ and $ \bigl... | The easiest way of computing the signs is to make them match; we know that sin x > 0 if 0 < x < π and that cos x > 0 if -π/2 < x < π/2. Knowing whether sin A is greater than 0 or less than zero tells you whether $\sqrt{1-\mathrm{sin} A}$ is greater or less than $\sqrt{1+\mathrm{sin} A}$; that in turn lets you figure o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Evaluate $\int \frac{1}{\sin x\cos x} dx $ Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However... | The second method gives the same answer as the first. By the first method, the answer you get is $-\log(\cos x) + \log(\sin x)$. The first term is the same as what you get by the second method.
What you need to show is that $\log(\sin x) = \frac{1}{2}\log(1-\cos x) + \frac{1}{2}\log(1+\cos x)$.
\begin{equation}
\begin{... | {
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"url": "https://math.stackexchange.com/questions/9075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 3
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Finding roots of the fourth degree polynomial: $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. My son is taking algebra and I'm a little rusty. Not using a calculator or the internet, how would you find the roots of $2x^4 + 3x^3 - 11x^2 - 9x + 15 = 0$. Please list step by step. Thanks, Brian
| Guessing one root sometimes opens up the whole equation for you.
First notice that $\displaystyle x=1$ gives 0. So $\displaystyle x-1$ is a factor.
Next, rewrite as
$\displaystyle 2x^4 - 2x^3 + 5x^3 - 5x^2 -6x^2 + 6x - 15x + 15$
This is to try and get $x-1$ as a factor.
This gives us
$\displaystyle 2x^3(x-1) + 5x^2(x-... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Factorize polynomial I am trying to factorize $-6x^5+15x^4-30x^2+30x-13$ for hours:( Could someone help me? I tried making a system of equations from $(Ax^3 + Bx^2 + Cx + D) (Ex^2 + Fx + G)$ but it is a nightmare:(
In case you are interested, the system is:
$AE = -6$
$AF + BE = 15$
$AG + BF + CE = 0$
$BG + CF + DE = -3... | First note that the domain over which the function makes sense in real variables is when $x^2 - 3x + 2 > 0$ i.e. when $(x-1)(x-2) > 0$ i.e. when $x > 2$ or $x < 1$. Now the way out is to rewrite
$\log \frac{x^2 - 3x + 2}{x^2 + 1}$ as $\log (x-1) + \log (x-2) - \log (x^2+1)$ if $x > 2$
and
$\log \frac{x^2 - 3x + 2}{x^2... | {
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Proving $(2n+1) + (2n+3) + \cdots + (4n-1) = 3n^{2}$ for all positive integers $n$
Prove $(2n+1) + (2n+3) + (2n+5) +
\cdots + (4n-1) = 3n^{2}$ for all
positive integers $n$.
So the provided solution avoids induction and makes use of the fact that $1 + 3 + 5 + \cdots + (2n-1) = n^{2}$ however I cannot understand th... | Your initial problem seems to be seeing
$$1 + 3 + 5 + \cdots + (4n-1) = 1 + 3 + 5 + \cdots + (2n-1) + (2n+1) + (2n+3) + \cdots + (4n-1)$$
so the LHS is $(2n)^2 = 4n^2$ and the left half of the RHS is $n^2$, leaving $3n^2$ for the right half of the RHS.
For induction, you need to start with one term where $(2 \times 1 +... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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How to solve $x^3 + 2x + 2 \equiv 0 \pmod{25}$? My attempt was:
$x^3 + 2x + 2 \equiv 0 \pmod{25}$
By inspection, we see that $x \equiv 1 \pmod{5}$. is a solution of $x^3 + 2x + 2 \equiv 0 \pmod{5}$. Let $x = 1 + 5k$, then we have:
$$(1 + 5k)^3 + 2(1 + 5k) + 2 \equiv 0 \pmod{25}$$
$$\Leftrightarrow 125k^3 + 75k^2 + 25k ... | This shows, that there is no modular zero for $x^3+2x+2\equiv 0 \; (\text{mod}\; 25)$ of the form $x = 1 + 5k$, since clearly $ 5 \not\equiv 0 \; (\text{mod}\; 25)$.
Now go ahead and check $x = 3 + 5k$, because $x \equiv 3 \; (\text{mod}\; 5)$ is the other solution to $x^3+2x+2\equiv 0 \; (\text{mod}\; 5)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Factorization of sum of two square The difference is $a^2 - b^2 = (a - b).(a + b)$
But what about when I have $a^{25} + 1$ ? According to wolfram alpha, the alternate form is:
*
*$(a+1) (a^4 -a^3 + a^2 -a + 1)( a^{20} - a^{15} + a^{10} -a^5 +1)$
However, the square root of 25 is a rational number 5.
But If I had 5... | If $n$ is odd then the polynomial $p(x)=x^n+1$ has a zero at $x=-1$, $p(-1)=(-1)^n+1=-1+1=0$. By the factor theorem, $x+1$ is therefore a factor of $p(x)$. Using polynomial long division (or synthetic division) you can show that $x^n+1=(x+1)(x^{n-1}-x^{n-2}+x^{n-3}-\cdots-x+1)$.
For example, consider $p(x)=x^5+1$. B... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
exponential equation $$\sqrt{(5+2\sqrt6)^x}+\sqrt{(5-2\sqrt6)^x}=10$$
So I have squared both sides and got:
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2\sqrt{1^x}=100$$
$$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2=100$$
I don't know what to do now
| $(\sqrt{(5+2\sqrt6)^x}+(\sqrt{(5-2\sqrt6)^x}=10$
We have, $5+2\sqrt6 = (\sqrt{2} + \sqrt{3})^2$ and $5 - 2\sqrt6 = (\sqrt3-\sqrt2)^2$
The equation $\Leftrightarrow \left(\sqrt{2} + \sqrt{3}\right)^x + \left(\sqrt3-\sqrt2\right)^x = 10$
Let $$t = \left(\sqrt3+\sqrt2\right)^x = \frac{1}{\left(\sqrt3-\sqrt2\right)^x}$$ wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/28157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 5
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Sum of cubed roots I need to calculate the sums
$$x_1^3 + x_2^3 + x_3^3$$
and
$$x_1^4 + x_2^4 + x_3^4$$
where $x_1, x_2, x_3$ are the roots of
$$x^3+2x^2+3x+4=0$$
using Viete's formulas.
I know that $x_1^2+x_2^2+x_3^2 = -2$, as I already calculated that, but I can't seem to get the cube of the roots. I've tried
$$(... | If $x_1,x_2,x_3$ are the roots of $x^3+2x^2+3x+4=0$ then $$x^3+2x^2+3x+4 = (x-x_1)(x-x_2)(x-x_3) $$ $$= x^3 - (x_1 + x_2 + x_3)x^2 + (x_1 x_2 + x_1 x_3 + x_2 x_3)x - x_1 x_2 x_3 = x^3 - e_1 x^2 + e_2 x - e_3.$$ So $e_1 = -2$, $e_2 = 3$ and $e_3 = -4$.
Now the trick is to express the power sums $x_1^3 + x_2^3 = x_3^3$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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How to find $x$ for $1 + \sin(x/2) = \cos x$? How to find $x$ for $1 + \sin(x/2) = \cos x$ ?
From the equation, I can figure out that it is satisfied at $x = 0$ by looking. How do I find the other solutions to this equation?
| Re-writing this as $\cos(2y) = 1 + \sin(y)$ for $y = \frac{x}{2}$ and using the fact that $\sin^2(y) = \frac{1 - \cos(2y)}{2}$, we can simplify the equation to $1 - 2\sin^2(y) = 1 + \sin(y)$ and thus $2\sin^2(y) + \sin(y) = (2\sin(y) + 1)\sin(y) = 0$. This means either $\sin(y) = 0$ or $\sin(y) = \frac{-1}{2}$, which h... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Integral of $\frac{1}{(1+x^2)^2}$ I am in the middle of a problem and having trouble integrating the following integral:
$$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$
I tried doing partial fractions and got:
$$1=A(1+x^2)+B(1+x^2)$$
I have no clue how to solve this since it is obvious there is no way to cancel out either $... | $$\frac{1}{(1+x^2)^2} = \left(\frac{1}{1+x^2}-\frac{x^2}{(1+x^2)^2}\right)= \left(\frac{1}{1+x^2}+\frac x2\left(\frac{1}{ 1+x^2 }\right)'\right)$$$$=\left(\frac{1}{1+x^2}+\left(\frac 12 \frac {x}{1+x^2}\right)'-\frac 12 \frac {1}{1+x^2}\right) = \frac 12 \left(\arctan x+\frac{x}{1+x^2}\right)'.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/35924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 10,
"answer_id": 3
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How to prove an alternating series is convergent? How to prove that the sequence $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{\sqrt{n}}$ is convergent? I was trying to find the upper bound and lower bound of the partial sum $s_k$ and use Squeeze Theorem to figure out the limit, but I couldn't find the lower bound for $s_k$... | This is an alternating series. Letting $a_n = \frac{1}{\sqrt{n}}$, for $n\geq 1$, then the series is $\sum (-1)^{n-1}a_n$. Notice that the sequence $a_n$ is strictly decreasing, as $a_1\gt a_2\gt a_3\gt\cdots$.
Now, consider the sequences of even and odd terms of the partial sum sequence:
$$\begin{align*}
s_1 &= 1\\
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Linear Algebra Question My question is; How can I find an equation relating $a,b$, and $c$ so that the linear system
$$\begin{cases}2x+y-z=a\\ x-2y-3z=b\\ -3x-y+2z=c\end{cases}$$
is consistent for any values of $a,b$, and $c$ that satisfy that equation.
Thanks,
| I don't know what tools you have available, but here is how I would consider this problem. If it uses terminology you do not know yet, you might come back to my answer later. This is intentionally not a direct path to the solution, rather the path I went though to solve the problem.
Write this system as a matrix equa... | {
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Explaining an algebra step in $ \frac{n^2(n+1)^2}{4} + (n+1)^3 = \frac{(n+1)^2}{4}(n^2+4n+4)$ I have encountered this step in my textbook and I do not understand it, could someone please list the intermediate steps?
$$ \frac{n^2(n+1)^2}{4} + (n+1)^3 = \frac{(n+1)^2}{4}(n^2+4n+4). $$
Thanks,
| Add the fractions, factor out common terms in the numerator and then rearrange:
$$
\displaystyle\; \; \frac{n^2(n^2 + 1)^2}{4} + (n + 1)^3
$$
$$
= \displaystyle\frac{n^2(n^2 + 1)^2}{4} + \frac{4(n + 1)^3}{4}
$$
$$
= \displaystyle\frac{n^2(n^2 + 1)^2 + 4(n+1)^3}{4}
$$
$$
= \displaystyle\frac{(n^2 + 1)^2(n^2 \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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How do I get the square root of a complex number? If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
| You can also do following (technique often advised at school) :
*
*Let's write $z² = 9 + 4i$ with $z = a + bi$. The goal is to find $z$
Thus we have $(a + bi)² = 9 + 4i$ and if you expend we get $a²+ 2abi - b² = 9 + 4i$
If you identify the real and imaginary parts, you obtain :
$a²-b² = 9$ (1)
and
$2ab= 4$ (... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "121",
"answer_count": 12,
"answer_id": 8
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Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:
$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
I really have no idea why this statement is true. Can someone please explain ... | Another take, similar to Euler's (?) proof given by @leonbloy. We know that if:
$\begin{align}
A(z)
&= \sum_{n \ge 0} a_n z^n
\end{align}$
then (writing $\mathtt{D}$ for the derivative):
$\begin{align}
z \mathtt{D} A(z)
&= \sum_{n \ge 0} n a_n z^n
\end{align}$
Also:
$\begin{align}
\frac{A(z)}{1 - z}
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "145",
"answer_count": 32,
"answer_id": 11
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Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$ Derive $\frac{d}{dx} \left[\sin^{-1} x\right] = \frac{1}{\sqrt{1-x^2}}$ (Hint: set $x = \sin y$ and use implicit differentiation)
So, I tried to use the hint and I got:
$x = \sin y$
$\frac{d}{dx}\left[x\right] = \sin y\frac{d}{dx}$
$\frac{dx}{dx} =... | First, note that $\displaystyle \sin^{-1}: [-1,1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]$
The range is important, as for this range, you have that if $y = \sin^{-1} x$ then $\cos y = \sqrt{1 - x^2}$, as we have $\sin y = x$ and $\cos y \ge 0$ whenever $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.
i.e. since $\sin^2 y + \cos^... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 4
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What is going on in this step? (from arc length problem) I am confused as to what is occurring in this step in an arc length problem:
Could anyone take a stab at trying to explain it to me? thanks
| First, the square root of the quotient is the quotient of the square roots:
$$\sqrt{\frac{x^{2/3}+1}{x^{2/3}}} = \frac{\sqrt{x^{2/3}+1}}{\sqrt{x^{2/3}}}.$$
Next, the square root in the denominator simplifies with the exponent, since:
$$\sqrt{x^{2/3}} = \left(x^{2/3}\right)^{1/2} = x^{1/3}.$$
Next, introduce a factor of... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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Help with integrating 101: $\int y \ln{y}\,\mathbb dy$ Would appreciate it if someone would please help me solve this
$$\int y\;\ln y\, \mathbb dy$$
taking time to explain reason for each step taken.
Thanks in advance!
| Here is a method of doing this by substitution. Put $y=e^{x}$. Then you have $dy = e^{x} \ dx$. Substituting we have
\begin{align*}
\int e^{2x} \cdot x \ dx &= \frac{1}{4}\int e^{t} \cdot t\ dt \qquad \Bigl[ \text{substituting} \ t = 2x \Bigr] \\ &= \frac{1}{4}\int e^{t} \cdot \bigl( t + 1 \bigr) \ dt - \frac{1}{4}\i... | {
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Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$ Find $\cos(x+y)$ if $\sin(x)+\sin(y)= a$ and $\cos(x)+\cos(y)= b$.
| An approach using complex numbers/geometry:
If $c = b + ia$, and if $z = \cos x + i \sin x$ and $z_1 = c - z = \cos y + i \sin y$, you are looking at the real part of $w = \cos(x+y) + i \sin (x+y) = zz_1 = z(c-z)$ with the restriction that $|z| = 1$ and $|z_1| = |c - z| = 1$.
Now the points satisfying $|z| = 1$ and $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 3
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Two problems on number theory I need some ideas (preferable some tricks) for solving these two problems:
Find the largest number $n$ such that $(2004!)!$ is divisible by
$((n!)!)!$
For which integer $n$ is $2^8 + 2^{11} + 2^n$ a perfect square?
For the second one the suggested solution is like this : $ 2^8 + 2^{11... | A way to do the second problem is the following. Check small values of $n<8$ by hand (nothing there). Then assume that $n\ge8$. Now $2^8+2^{11}+2^n=2^8(1+8+2^{n-8})$ is a perfect square, iff the latter factor $9+2^{n-8}$ is a perfect square also. But
$$
9+2^{n-8}=m^2\Leftrightarrow 2^{n-8}=m^2-9=(m-3)(m+3).
$$
So by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/57177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
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Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$
If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$.
How to solve this one?
(Here's the source of inspiration for the problem.)
| You can also use the method of Lagrange multipliers to maximize/minimize $f(a,b,c)=a+b+c$ given $g(a,b,c)=a^2+b^2+c^2=27$. We need the gradient of $f$ to be a multiple of the gradient of $g$, i.e.,
$$1=\lambda 2a,\quad 1=\lambda 2b,\quad 1=\lambda 2c,$$
where $\lambda$ is some real number. Hence:
$$\lambda = \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/59185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
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Obtaining Differential Equations from Functions I can now recognise the order and the type of differential equations.Let's say
$$\frac{dy}{dx} = x^2 - 1$$
is a first order ODE,
$$\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 + y = 0$$
is a second order ODE and so on. I am having trouble to obtain a differentia... | Eliminate $a$ and $b$ from $$y = ae^{2x} +be^{3x}\tag{1}$$
$$\frac{dy}{dx} = 2ae^{2x} + 3be^{3x}\tag{2}$$
$$d^2y/dx^2 = 4ae^{2x} + 9be^{3x}\tag{3}$$
$(1)\cdot 2 -(2)$ we get
$$2y - \frac{dy}{dx} = -be^3x$$
$$b = e^{-3x} \left(\frac{dy}{dx} - 2y\right)\tag{4}$$
$(2)*2 - (3)$
$$2 \frac{dy}{dx} - \frac{d^2y}{dx^2} = - 3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Olympiad Inequality Problem Consider three positive reals $x,y,z$ such that $xyz=1$.
How would one go about proving:
$$\frac{x^5y^5}{x^2+y^2}+\frac{y^5z^5}{y^2+z^2}+\frac{x^5z^5}{x^2+z^2}\ge \frac{3}{2}$$
I really dont know even where to begin! It looks a BIT like Nesbitts? Maybe?
| By Holder $$\sum_{cyc}\frac{x^5y^5}{x^2+y^2}\geq\frac{(xy+xz+yz)^5}{(1+1+1)^3\sum\limits_{cyc}(x^2+y^2)}=\frac{(xy+xz+yz)^5}{54(x^2+y^2+z^2)}.$$
Hence, it remains to prove that $$(xy+xz+yz)^5\geq81(x^2+y^2+z^2).$$
Let $x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Thus, since $xy+xz+yz$ and $x^2+y^2+z^2$ does not depend on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 3
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Solve equation: $\frac{7}{\sqrt{343^{5x-1}}} = 2401^{-6.7}$ I need help to solve this equation, thanks in advance.
$$\frac{7}{\sqrt{343^{5x-1}}} = 2401^{-6.7}$$
| $$\frac{7}{\sqrt{343^{5x-1}}} = 2401^{-6.7}$$
Since $343=7^3$ and $2401 = 7^4$, we can write
$$
\frac{7}{\sqrt{7^{3(5x-1)}}} = 7^{4\cdot(-6.7)}
$$
and then
$$
\frac{7}{\left(7^{3(5x-1)}\right)^{1/2}} = 7^{4\cdot(-6.7)}
$$
So
$$
7^{1 - (1/2)(3)(5x-1)} = 7^{4\cdot(-6.7)}.
$$
Hence
$$
1 - \frac12 \cdot3(5x-1) = 4\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/62022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
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Solving $8=x(2(1-\sqrt{5}))+(1-x)(2(1+\sqrt{5}))$ I came up with this equation during my homework :
$8=x(2(1-\sqrt{5}))+(1-x)(2(1+\sqrt{5}))$
My algebra is weak and I can't seem to find a way to solve for x nicely
Could someone please show me a decent way of doing this?
Thanks alot, Jason
| $$8 = x\left [ 2-2\sqrt{5} \right ] + \left ( 1 - x \right )\left [ 2 + 2\sqrt{5} \right ]$$
use foil
$$8 = 2x - 2x\sqrt{5} + 2 + 2\sqrt{5} - 2x - 2x\sqrt{5}$$
$$8 = -4x\sqrt{5} + 2\sqrt{5} + 2$$
subtract 2 and square both sides
$$36 = 16x^25 + 20$$
subtract 20 from both sides to obtain
$$80x^2 = 16$$
Now divide both ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/62073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How to solve this equation: $\frac{\sqrt[5]{x^3 \sqrt{x\sqrt[3]{x^{-2}}}}}{\sqrt[4]{x\sqrt[3]{x}}}=3$ Please, help me to solve this equation:
$$\frac{\sqrt[5]{x^3\sqrt{x\sqrt[3]{x^{-2}}}}}{\sqrt[4]{x\sqrt[3]{x}}}=3$$
I tried to shorten fraction, but I get very weird numbers like
$$\frac{\sqrt[30]{x^{19}}}{\sqrt[3]{x}}... | You're almost done. Note that $\frac{1}{x} = x^{-1}$, so
$$\frac{\sqrt[30]{x^{19}}}{\sqrt[3]{x}} = \sqrt[30]{x^{19}}(\sqrt[3]{x})^{-1} = x^{19/30}x^{-1/3} = x^{9/30}$$
So then
$$x^{9/30} = 3 \quad \Longleftrightarrow \quad x = 3^{30/9} = 3^{10/3} = \sqrt[3]{3^{10}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/63686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Inverse Laplace Transform -s domain How can I find the inverse Laplace transforms of the following function?
$$ G\left(s\right)=\frac{2(s+1)}{s(s^2+s+2)} $$
I solved so far. After that, how do I do?
$$ \frac{1}{s}+\frac{1}{s^2+s+2}+\frac{s}{s^2+s+2}=G\left( s \right)$$
| To find the inverse Laplace transforms of the function $\ G\left(s\right)=\dfrac{2\left(s+1\right)}{s\left(s^2+s+2\right)} $
You have solved up to partial fraction form of $G\left(s\right)$ i.e
$$G\left(s\right)=\frac{1}{s}+\frac{1}{s^2+s+2}+\frac{s}{s^2+s+2}$$
Now taking the Laplace inverse $$\begin{align}\mathcal{L^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Parametric form of an ellipse given by $ax^2 + by^2 + cxy = d$ If $c = 0$, the parametric form is obviously $x = \sqrt{\frac{d}{a}} \cos(t), y = \sqrt{\frac{d}{b}} \sin(t)$.
When $c \neq 0$ the sine and cosine should be phase shifted from each other. How do I find the angular shift and from there how do I adjust the f... | You would complete squares: $\left(a x + \frac{1}{2} c y\right)^2 + \left(a b - \frac{c^2}{4} \right) y^2 = a d$.
From there: $a x + \frac{c}{2} y = \sqrt{a d} \sin(t)$ and $\sqrt{a b - \frac{c^2}{4}} y = \sqrt{a d} \cos(t)$, assuming $c^2 < 4 a b$, and $a d > 0$.
Solving for $x$ and $y$ and denoting $\mathcal{D} = 4 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Explicit solution to $100(y + \frac{100}{10-y}) = x + 1000$ $$
100\left(y + \frac{100}{10-y}\right) = x + 1000
$$
I'm looking to have the equation in the form $y(x) = ...$ Seems pretty simple, but I'm just not seeing it.
The answer is:
$$
y(x) = \frac{1}{200}\sqrt{x^2 + 4000x} − x
$$
Wolfram|Alpha returns a slightl... | Wolfram Alpha is correct. Starting with
$$100\left(y + \frac{100}{10-y}\right) = x + 1000$$
and dividing both sides by $100$:
$$y+\frac{100}{10-y}=\frac{x}{100}+10$$
multiplying both sides by $(10-y)$:
$$-y^2+10y+100=\left(\frac{x}{100}+10\right)(10-y)$$
collecting everything on one side:
$$y^2-\left(20+\frac{x}{100}\r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/71453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Complex numbers, solutions of $1-z+z^2=0$ $$z_1 \text{ and } z_2 \text{ are the solutions of } 1-z+z^2=0$$
$$E=(z_1^4-z_1^3+2z_1^2-2z_1+1)^{2005}+(z_2^4-z_2^3+2z_2^2-2z_2+1)^{2005}$$
Which is the value of $E$ ?
I have solved the equation:
\begin{align*}\Delta = 1-4=-3=3i^2&\Rightarrow
z_{1,2}=\frac{1\pm i\sqrt{3}}{2}
\... | If you know that $z_i^2-z_i+1 = 0$, then since
$$x^4 - x^3 + 2x^2 - 2x + 1 = (x^2-x+1)(x^2+1) - x,$$
it follows that:
$$z_1^4 - z_1^3 + 2z_1^2 - 2z_1 + 1 = (z_1^2 - z_1 + 1)(z_1^2 + 1) - z_1 = 0(z_1^2+1) - z_1 = -z_1.$$
Much simpler than trying to work directly with the roots.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/76392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Maximum Value Question I have a question about this following question:
Let $a>0$. Show that the maximum value of $f(x):=\frac{1}{1+|x|}+\frac{1}{1+|x-a|}$ is $\frac{2+a}{1+a}$
I am wondering if I am headed in the right direction with the following process. I first adress the issue of the absolute values by rewriti... | This was a response to the question $f(x):=\frac{1}{1+|x|}-\frac{1}{1+|x-a|}$
Perhaps I am misunderstanding something, but doesn't the graph look something like this?
If $x \le 0 $ then $f(x)=\frac{1}{1+|x|}-\frac{1}{1+|x-a|}=\frac{1}{1-x}-\frac{1}{1+a-x} =\frac{a}{(1-x)(1+a-x)}$ which is an positive increasing functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing whether two numbers are equal or not $\dfrac{\sin (2x+y)}{\sin (2x)} =\dfrac{\sin (x+2y)}{\sin (2y)}$,where $0<x,y\le\dfrac{\pi}{4}$ .
Can I show that $x=y $ or find two numbers $x,y$ such that $x\not=y$?
| Edited: I think I got it...there was typo in the first try.
After cross-multiplication, we get
$[2\sin y\sin(2x+y)]\cos y-[2\sin x\sin(x+2y)]\cos x=0$
$\Rightarrow[\cos(2x)-\cos2(x+y)]\cos y-[\cos(2y)-\cos2(x+y)]\cos x=0$
$\Rightarrow\cos2(x+y)[\cos x-\cos y]+[(2\cos^2x-1)\cos y-(2\cos^2y-1)\cos x]=0$
$\Rightarrow[\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Modular exponentiation by hand ($a^b\bmod c$) How do I efficiently compute $a^b\bmod c$:
*
*When $b$ is huge, for instance $5^{844325}\bmod 21$?
*When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$?
*When $(a,c)\ne1$, for instance $6^{103... | Wikipage on modular arithmetic is not bad.
*
*When $b$ is huge, and $a$ and $c$ are coprime, Euler's theorem applies:
$$
a^b \equiv a^{b \, \bmod \, \phi(c)} \, \bmod c
$$
For the example at hand, $\phi(21) = \phi(3) \times \phi(7) = 2 \times 6 = 12$.
$$
\Rightarrow 844325 \bmod 12 = 5,\ \text{so}\ 5^5 = 5 \time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "128",
"answer_count": 11,
"answer_id": 1
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Proving inequality $x^{10}-x^6+x^2-x+1>0$ How can the inequality $x^{10}-x^6+x^2-x+1>0$ be proved
a) using elementary mathematical methods？
b) using higher mathematical methods？
| Another elementary method:
We want to show that $x^{10}+x^2+1 >x^6+x$. If we can show it for $x \ge 0$, it will be true always. This is because if $x$ is positive, replacing $x$ by $-x$ does not change $x^{10}+x^2+1$, but turns $x^6+x$ into the smaller $x^6-x$.
We now take care of $x \ge 0$. If $x\ge 1$, then $x^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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A Recurrence Relation Problem
In a standard elimination tournament, a player wins $\$100k$ when she/he wins a match in the $k$th round. Develop and solve a recurrence relation for $a_n$, the total amount of money given away in a tournament with $n$ entrants, where $n$ is assumed to be a power of $2$.
I seem always to... | I think Ross's answer is the beginning of things but that he had it wrong. Let me explain.
You have $2^n$ players to begin with, hence $2^{n-1}$ matches for the first round. This gives us $2^{n-1}$ winners for the first round, hence $1 \times 100 \times 2^{n-1}$ dollars given away. Then $2^{n-1}$ players go to the seco... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that there are infinitely many reducible polynomials of the form $x^n+x+1$ in $\mathbf{F}_2[x]$ Here is a question from an old exam:
Show that there are infinite $n\in \mathbf{N}, A= x^{n}+x+1 $ which are reducible over $\mathbf{F}_{2}[x]$.
Using André Nicolas' and Qiaochu Yuan's hint: $x^{2}+x+1$ as dividing ... | Proof by induction.
Base case ($n=0$):
$$\begin{align}
(x^2+x+1)\left(x^3+\sum\limits_{i=0}^0 (x^{3i}+x^{3i+2})\right)&=(x^2+x+1)(x^3+x^2+1)\\&=x^5+2x^4+2x^3+2x^2+x+1\\&=x^5+x+1=x^{3(0)+5}+x+1
\end{align}$$ (working in $\mathbb{F}_2[x]$).
Inductive hypothesis ($n \geq 0$):
Suppose that $$(x^2+x+1)\left(x^{3(n+1)}+\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/82859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one? I came across a question that asked me to divide $-2x^3+4x^2-3x+5$ by $4x^2+5$. Can anyone help me?
| The answer in no way depends on whether $a$ and $b$ are more than $1$ or less; the only fact about those that you need is that the polynomial you're dividing by is not zero. You have this:
$$
\begin{array}{ccccccccccccccccc}
\\
4x^2+5 & \big) & -2x^3 & + & 4x^2 & - & 3x & + & 5 \\
& &
\end{array}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Solve $a, b$ in the improper integral Find $a, b \in\mathbb{R}$ such that $$\int^\infty_1\left(\dfrac{2x^2+bx+a}{x(2x+a)}-1\right) \mathrm{d}x=1$$
| $$I = \int_1^{\infty} \left(\frac{bx+a-ax}{x(2x+a)} \right) dx$$ $$ \left(\frac{bx+a-ax}{x(2x+a)} \right) = \frac{1}{x} + \frac{b-a-2}{2x+a}$$ $$I = \int_1^{\infty} \left(\frac{1}{x} + \frac{b-a-2}{2x+a} \right) dx = \left[ \log(x) + \frac{b-a-2}{2} \log \left(x + \frac{a}{2} \right)\right]_1^{\infty}$$ Since, we get a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to get closed form from generating function? I have this generating function:
$$\frac{1}{2}\, \left( {\frac {1}{\sqrt {1-4\,z}}}-1 \right) \left( \,{
\frac {1-\sqrt {1-4\,z}}{2z}}-1 \right)$$
and I know that $\frac {1}{\sqrt {1-4\,z}}$ is the generating function for the sequence $\binom {2n} {n}$, and $\frac {1-\... | It’s not necessary to use the generalized binomial theorem and the gamma function. Let $$g(x)=\frac12\left(\frac1{\sqrt{1-4x}}-1\right) \left(\frac{1-\sqrt{1-4x}}{2x}-1\right)$$ and $u=\sqrt{1-4x}$. Then
$$\begin{align*}
g(x)&= \frac12\left(\frac1u-1\right)\left(\frac{1-u}{2x}-1\right)\\
&=\frac12\left(\frac{1-u}{2xu}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to prove :If $p$ is prime greater than $3$ and $\gcd(a,24\cdot p)=1$ then $a^{p-1} \equiv 1 \pmod {24\cdot p}$? I want to prove following statement :
If $p$ is a prime number greater than $3$ and $\gcd(a,24\cdot p)=1$ then :
$a^{p-1} \equiv 1 \pmod {24\cdot p}$
Here is my attempt :
The Euler's totient function ca... | Since $(a,24\cdot p)=1$, it also follows that $(a,p)=(a,3)=(a,8)=1$.
By the generalization of Fermat's little theorem, $a^{p-1}\equiv 1\pmod{p}$, $a^2\equiv 1\pmod{3}$, and $a^4\equiv 1\pmod{8}$. But $a^4\equiv 1\pmod{8}$, implies $a\equiv 1,3,5,7\pmod{8}$, and in all cases $a^2\equiv 1\pmod{8}$.
Since $p-1$ is even,... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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How to find the equation of the tangent line to $y=x^2+2x-4$ at $x=2$?
I'm given a curve $$y=x^2+2x-4$$
How do I find the tangent line to this curve at $x = 2$?
| Here's an algebraic approach that avoids the explicit use of derivatives.
We are given a quadratic function $f(x) = x^2 + 2x -4$, and we want to find the equation of the tangent to the parabola $y = f(x)$ at the point $(2, 4)$. (Note that $f(2) = 2^2 + 2 \cdot 2 - 4 = 4$.) Assume that it is given by the equation
$$
y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
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Sangaku: Show line segment is perpendicular to diameter of container circle "From a 1803 Sangaku found in Gumma Prefecture. The base of an isosceles triangle sits on a diameter of the large circle. This diameter also bisects the circle on the left, which is inscribed so that it just touches the inside of the container ... | For those who do not like inversion... =P
Let $A$ be the big circle with centre $O$ and diameter $PQ$
Let $B$ be the circle internally tangent to $A$ at $P$ and intersecting $PQ$ again at $M$ and having centre $J$
Let $R$ be on $A$ such that $\overline{MR} = \overline{QR}$
Let $C$ be the circle that is tangent to $A$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 1
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Very curious properties of ordered partitions relating to Fibonacci numbers I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon.
We call an ordered Partition of a positive integer $n$ as the way of wri... | This problem has a solution using ordinary generating functions.
First question.
Observe that
$$\sum_{q\ge 0} q z^q = \frac{z}{(1-z)^2}.$$
Therefore the generating function of the contribution from partitions
with $k$ terms is given by
$$\left(\frac{z}{(1-z)^2}\right)^{k-1} \frac{z}{1-z}$$
and the contribution from a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 3,
"answer_id": 1
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How to use modulo to find the last character of an exponentiation? I already found interesting answers in other questions to this topic. Yet, I still don't get it well enough to use it. Can someone please show it with an example? I won't tell you mine, because it's a homework and I really want to understand it myself.
| For $a^N\equiv x\pmod{10}$ (with $0\leq x<10$), a general strategy is to start finding a small exponent $n$ such that $a^n\equiv 0,1,-1\pmod{10}$.
Example 1:
$3^{1000}\equiv x\pmod{10}$. We have $3^2\equiv -1\pmod{10}$ thus $3^4=(3^2)^2\equiv (-1)^2\equiv 1\pmod{10}$, then $3^{1000}=(3^4)^{250}\equiv 1^{250}\equiv 1\p... | {
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"timestamp": "2023-03-29T00:00:00",
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Factoring polynomial with complex coefficients
Given the equation $z^2+4iz-13=0$, solve for $z$ without the quadratic formula.
In real numbers set, when I find this kind of equations I usually complete the perfect square trinomial.In this case:
$(z^2+4iz-4)-13+4=0$
$(z+2i)^2-9=0$
I chosen $-4$ because the number whos... | What you did is correct.
You can check by substitution: If $z=3-2i$ then
$$(z+2i)^2-9= ((3-2i)+2i)^2-9 = 3^2-9=0.$$
If $z=-3-2i$ then
$$
(z+2i)^2 - 9 = ((-3-2i) + 2i)^2 - 9 = (-3)^2 - 9 = 0.
$$
| {
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"url": "https://math.stackexchange.com/questions/107635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Given that $[n(n+1)(n+2)]^2 = 303916253\square96$, find the value of $\square$. Given that $[n(n+1)(n+2)]^2 = 303916253\square96$, find the value of $\square$.
Given that $[n(n+1)(n+2)]^2 = 30391625\square796$, find the value of $\square$.
Problem
Given that $[n(n+1)(n+2)]^2 = 3039162537\square6$, find the value of $\s... | Then, there's brute force:
303,916,253,?96 ~= 30.3 e+10 ~= (5.5 e+5) squared ~= (n cubed) squared.
So, 80 < n < 90 since 80 cubed = 512,000 and 90 cubed = 729,000.
Since 303,916,253,?96 ends with 6, the number that is squared must end with 4 or 6. Hence, none of n, n+1, and n+2 can have 0 or 5 in the one's digit.
This... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Cumulative distribution function question
Possible Duplicate:
Normal random variable $X$ and the cdf of $Y=aX+b$
I'm given a standard random variable $X$, and $Y = aX + b$:
How can I find the cumulative distribution function for Y as an integral of $f(x)=(\frac{1}{\sqrt{2}\pi\sigma}e^{-\frac{(x-u)^2}{\sigma^2}}$?
I ... | If $a>0$ then
$$
\begin{align}
& f_Y(y) = \frac{d}{dy} F_Y(y) = \frac{d}{dy}\Pr(Y\le y) = \frac{d}{dy}\Pr(aX+b\le y) = \frac{d}{dy}\Pr\left(X \le \frac{y-b}{a}\right) \\ \\
& = \frac{d}{dy}F_X\left( \frac{y-b}{a} \right) = f_X\left(\frac{y-b}{a}\right) \cdot \frac{d}{dy} \frac{y-b}{a} = \frac 1 a f_X\left(\frac{y-b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What is this formula? There is some formula that I can't precisely remember for polynomials, which goes something like $x^n-1 = (x-1)(\text{a lot of stuff})$. It could be more general, like $x^n - k$, or maybe it is just for the same powers, so $x^m - k^m$, but I think it's not just for the same powers. Does anyone kno... | What that formula is stating is that, when $x^n-1$ is dividided through $x-1$ one gets $$1+x+x^2+x^3+\cdots+x^{n-1}$$
This is derived in several ways. One is
Let $$S = 1+x+x^2+x^3+\cdots+x^{n-1}$$
Then one has that
$$Sx= x+x^2+x^3+\cdots+x^{n}$$
Thus substracting this two equation one gets
$$Sx-S = x^n-1$$
or
$$S(x-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
the cube of integer can be written as the difference of two square This Exercise $4$, page 7, from Burton's book Elementary Number Theory.
Prove that the cube of any integer can be written as the difference of two squares. [Hint: Notice that $n^{3}=(1^{3}+2^{3}+\cdots+n^{3})-(1^{3}+2^{3}+\cdots+(n-1)^{3}).$]
Is there a... | A more natural approach is to work out exactly which integers can be written as the difference of two squares, and then notice that all cubes are such.
So suppose $n=a^2-b^2$, with $n$, $a$ and $b$ all integers. Then we have $n=(a+b)(a-b)$. Setting $x=a+b$, $y=a-b$ we have $n=xy$ with $a=\frac{x+y}{2}$, $b=\frac{x-y}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
Fibonacci's final digits cycle every 60 numbers How would you go about to prove that the final digits of the Fibonacci numbers recur after a cycle of 60?
References:
The sequence of final digits in Fibonacci numbers repeats in cycles of 60. The last two digits repeat in 300, the last three in 1500, the last four in , ... | Note that
$$
\begin{pmatrix} F_{n+1}\\ F_{n+2} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} F_n \\ F_{n+1} \end{pmatrix}
$$
and
$$
\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}^{60} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \mod 10.
$$
One can verify that $60$ is the smalles... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 0
} |
Intuitive explanation for a polynomial expansion? Is there an ituitive explanation for the formula:
$$
\frac{1}{\left(1-x\right)^{k+1}}=\sum_{n=0}^{\infty}\left(\begin{array}{c}
n+k\\
n
\end{array}\right)x^{n}
$$
?
Taylor expansion around x=0
:
$$
\frac{1}{1-x}=1+x+x^{2}+x^{3}+...
$$
differentiate this k
t... | If by the binomial law you mean
$$
(1+x)^n=\sum_k\binom{n}{k}x^k\tag{1}
$$
then yes. Note that
$$
\binom{n}{k}=\frac{n(n-1)(n-2)\dots(n-k+1)}{k!}\tag{2}
$$
Consider what $(2)$ looks like for a negative exponent, $-n$:
$$
\begin{align}
\binom{-n}{k}
&=\frac{-n(-n-1)(-n-2)\dots(-n-k+1)}{k!}\\
&=(-1)^k\frac{(n+k-1)(n+k-2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$. Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$
I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyo... | Suppose $xy\mid x^2+y^2+1$ and let $t=\displaystyle\frac{x^2+y^2+1}{xy}$ such that $t\in\mathbb{N}$.
Construct the set,
$$S=\left\{(x,y)\in\mathbb{N}\times\mathbb{N} : \frac{x^2+y^2+1}{xy}=t\in\mathbb{N}\right\}$$
We deduce that $\displaystyle\frac{x^2+y^2+1}{xy} \ge 3$ because $\displaystyle\frac{x^2+y^2+1}{xy}<3$ imp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 5,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.