Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Sove the recurrence relation $a_n = 5a_{n-1}-8a_{n-2}+4a_{n-3}$ and $a_0 = 0$, $a_1 = 0$, $a_2 = 1$
Consider the recurrence relation $a_n = 5a_{n-1}-8a_{n-2}+4a_{n-3}$ and $a_0 = 0$, $a_1 = 0$, $a_2 = 1$. Find a closed form expression for $a_n$.
I have computed the OGF $A(z)$ of $(a_n)_n$ as
$$A(z) = \frac{z^2}{1-5z+... | Starting from
$$\frac{z^2}{1-5z+8z^2-4z^3}$$
then
\begin{align}
\sum_{n=0}^{\infty} a_{n} \, x^n &= \frac{x^2}{1-5x+8x^2-4x^3} = \frac{x^2}{(1-x)(1-2 x)^2} \\
&= \frac{1}{1 - x} - \frac{3}{2 (1 - 2 x)} + \frac{1}{2 \, (1 - 2 x)^2} \\
&= \sum_{n} x^n - \frac{3}{2} \, \sum_{n} 2^n \, x^n + \frac{1}{2} \, \sum_{n} 2^n (n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4594786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Attempt to solve $\lim_{x\to +\infty} \exp{\left(\frac{x^2}{1+x}\right)} - \left(1+\frac1x\right)^{x^2}$ I tried to solve this limit. Is it correct? There exists a more straightforward way?
$$\lim_{x\to +\infty} \exp{\left(\frac{x^2}{1+x}\right)} - \left(1+\frac1x\right)^{x^2}$$
$$\lim_{x\to +\infty} \exp{\left(\frac{x... | You are correct. A slightly shorter way with a more general result: let $a$ be a real number then, as $x\to +\infty$,
$$\begin{align}
\exp{\left(\frac{x^2}{a+x}\right)}& - \left(1+\frac1x\right)^{x^2}=
\exp{\left(\frac{x}{1+\frac{a}{x}}\right)}-\exp{\left(x^2\ln\left(1+\frac{1}{x}\right)\right)}\\
&=\exp{\left(x\left(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4595816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
In any triangle $\triangle ABC$, show that $4R\sin(\frac{A}{2})\sin(\frac{B}{2})\sin(\frac{C}{2})=r$ This is a problem problem I found in a JEE examination prep textbook, it was a "starred" question which I believe implies that it is more challenging than usual. It goes as follows:
In any triangle $\triangle ABC$, sho... | By using the area formula
$$\Delta=\sqrt{s(s-a)bc}\sin{\frac{A}{2}},$$
at each vertex of the triangle, we can have
$$\Delta^6=s^3(s-a)(s-b)(s-c)(abc)^2\sin^2{\frac{A}{2}}\sin^2{\frac{B}{2}}\sin^2{\frac{C}{2}}.$$
After simplifying by Heron's formula,
$$\Delta^4=s^2(abc)^2\sin^2{\frac{A}{2}}\sin^2{\frac{B}{2}}\sin^2{\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4597929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
find all positive integers that are the cube of the sum of their digits
(HMMT 2000 Guts #5). Find all positive integers that are the cube of the sum of their digits.
I've found a (very) tedious way to resolve the question described below. The tedious part is highlighted in a similar way to the question at the top.
... | Let $n=k^3.$ If $n$ has $6$ digits, the largest it could be is $(6\cdot 9)^3 = 157464$, which shows that one digit must be $1$. So in fact the largest it could be is $(1+5\cdot 9)^3 = 97336$ which is too small. So there are no $6$ digit (or larger) solutions. (I think this is the gist of @dezdichado's comment.)
Be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4598986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is there another way of solving this binomial problem? The problem is as follows: in $$(1+x)^n$$ find $n$ such that the fifth coefficient is $70$ and the seventh coefficient equals $28$.
I put $$ \binom{n}{4}=70 $$ $$\binom{n}{6}=28 $$
I ended up with two polynomial equations:
$$ n^4-6n^3+11n^2-6n-1680=0 $$
$$ n^6-15... | Divide $\binom{n}{4}$ by $\binom{n}{6}$, it will give us
$$
\frac{70}{28} = \frac{\binom{n}{4}}{\binom{n}{6}} = \frac{\frac{n!}{4!\left(n-4\right)!}}{\frac{n!}{6!\left(n-6\right)!}} = \frac{6!\left(n-6\right)!}{4!\left(n-4\right)!} = \frac{6\cdot5\cdot4!\left(n-6\right)!}{4!\left(n-4\right)\cdot\left(n-5\right)\cdot\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4601559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Deflation of eigenvalue Could you explain to me the deflation?
For example if we have the matrix $A=\begin{pmatrix}1 & -0.5 & -1.5\\ -15 & -2.5 & 4.5\\ -15 & -4.5 & 2.5\end{pmatrix}$ how do we apply the deflation of the eigenvalue $\lambda_1=4$ ?
$$$$
EDIT :
I have done the following :
We have the eigenvalue $\lambda_... | We have $v_1=\begin{pmatrix}-1\\3\\1\end{pmatrix}$.
Let $P$ be the permutation matrix that swaps the first 2 elements.
Then $Pv_1=v_1'=\begin{pmatrix}3\\-1\\1\end{pmatrix}$ has the element with the largest magnitude first.
Let $R$ be the matrix of row operations that transforms $v_1'$ into $e_1$. That is $Rv_1'=e_1$.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4602697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
On generalizing $\frac{17-\sqrt{17}}8 =\sin^2(t)+\sin^2(2t)+\sin^2(4t)+\sin^2(8t)$? Given $\color{blue}{t = 2\pi/p}$ for the appropriate prime $p=4m+1$.
I. Sine
$$\begin{align}
\frac{5+\sqrt{5}}8 &=\sin^2(t)\\
\frac{13+\sqrt{13}}8 &=\sin^2(t)+\sin^2(3t)+\sin^2(4t)\\
\frac{17+\sqrt{17}}8 &=\sin^2(3t)+\sin^2(5t)+\sin^2(... | I found an algorithm to derive the correct set $H$ of $a_k$ for any prime $\ p=4m+1 \ $ (or its complement).
$H=\{ 1\}$.
For $k$ from $2$ to $2m$:
*
*Compute $\alpha$ such that $\ \ k^2 \equiv \alpha \pmod{4m+1}\ \ $ and $\ \ 1\leqslant \alpha \leqslant 4m \ \ $
*if $\ 1\leqslant \alpha \leqslant 2m \ $ then $\ \be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4603836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Find the area of Quadrilateral $ABCD$. A puzzle for 10th graders. As the title suggests, the problem in this post was meant to be a puzzle for 10th graders, so claims the person who posted this on a language exchange platform: The problem is as follows:
Given a Quadrilateral $ABCD$ with internal point $P$, where $AP=1... | The problem is described as
As the title suggests, the problem in this post was meant to be a puzzle for 10th graders,
but I imagine for very mathematically talented students. I
simplify the original problem into an equivalent form.
Suppose a square $ABCE$ is such that the side lengths of $AB$
and $CE$ are equal to $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4604319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Proof of $a<\frac{1}{a}I am currently working on Velleman's How to Prove It section 3.2, question nine (question eight in the second edition).
Why does he hint in the solution section to go through the steps to first prove $a<1$ and $a<0$, rather than just using a proof by contradiction by assuming $a\ge-1$ to conclude... | Suppose $a$ and $b$ are nonzero real numbers such that $a < \frac{1}{a} < b < \frac{1}{b}$. We want to show that $a < -1$.
To this end, suppose towards a contradiction that $a \geq -1$. Then there are three cases to consider:
Case 1: Suppose that $-1 \leq a < 0$. Then since $a$ is negative, we know that $-a$ is positiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4606843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Where did I go wrong with this integration? $$\int_0^{\frac {1}{\sqrt 3}} \frac {1}{\sqrt {2-3x^2}}$$
$$\int_0^{\frac {1}{\sqrt 3}} \frac {1}{\sqrt {2(1-\frac32x^2)}}$$
$$\frac 1{\sqrt 2}\int_0^{\frac {1}{\sqrt 3}} \frac {1}{\sqrt {1-\frac32x^2}}$$
$$\bigg(\frac 1{\sqrt 2}\sin^{-1}{\sqrt {\frac 32 x}}\bigg)\bigg|_0^\fr... | We use $f'(x)=\frac{1}{\sqrt{1-x^2}}$ for $f(x)=\sin^{-1}(x)$ and $x\in(-1,1)$. Using the chain rule gives $f'(x)=\frac{a}{\sqrt{1-(ax)^2}}$ for $f(x)=\sin^{-1}(ax)$. Hence, we have
\begin{align*}
\int_{0}^{1/\sqrt{3}}\frac{1}{\sqrt{2-3x^2}}\mathrm dx
&=\sqrt{\frac{2}{2\cdot 3}}\int_{0}^{1/\sqrt{3}}\frac{\sqrt{\frac{3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4607606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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$\mathbb{Q}(\sqrt{-23}) \subseteq \mathbb{Q}(\zeta_{23})$ I want to prove $\mathbb{Q}(\sqrt{-23}) \subseteq \mathbb{Q}(\zeta_{23})$ using the hint from "Introduction to Cyclotomic Fields" Exercise 2.1. $\zeta_{23}$ is the primitive 23-th root of unity.
I know that $p=23$ is the only prime ramified in $\mathbb{Q}(\zeta_... | also, if $x^{23} = 1$ but $x \neq 1,$ take
$$z = x + x^2 + x^3 + x^4 + x^6 + x^8 +x^9 + x^{12} + x^{13} + x^{16} + x^{18} $$
Then
$$ z^2 + z + 6 = 0 $$
because gathering the same exponents mod 23 comes out to
$$z =6 \left( 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +x^7 + x^8 +x^9 + x^{10} + x^{11} +x^{12} + x^{13} +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4610537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Area of isosceles triangle I need to solve following problem:
The vertex of the isosceles triangle $ABC$ is the point $A(-1, 0)$, and the vertices $B$ and $C$ belong to the parabola $y^2 = 4x$. If the point $(0, 0)$ is the orthocenter of triangle $ABC$, then its area is equal to ?
I've tried to use fact that points C... | Given that the line connecting $B$ to the origin is $y = mx$, $y^2 = 4x$ gives $m^2 x^2 - 4x = 0$ $ \implies x(m^2 x - 4) = 0$, so the other intersections apart from $(0, 0)$ are when $x = \frac{4}{m^2}$, and $y = \pm \frac{4}{m}$.
Now finding the equation of the perpendicular which must pass through $(-1, 0)$, we have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4613968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
kernel of matrix in given basis Linear transformation $f$ is given in basis ${v_1, v_2, v_3}$ by matrix $M$:
$$\begin{pmatrix}0 & 2 & -1\\ -2 & 5 & -2\\ -4 & 8 & -3\end{pmatrix}.$$
Find kernel and image of matrix in terms of basis ${v_1, v_2, v_3}$
Okay so in order to find kernel I transform matrix using elementary row... | Consider the matrix representation of linear transformation $f$ in the ordered basis $ \beta=\{v_{1},v_{2},v_{3}\}$ given by $$[f]_{\beta}=\begin{pmatrix}0 & 2 & -1\\ -2 & 5 & -2\\ -4 & 8 & -3\end{pmatrix}$$
By definition, the image of $[f]_{\beta}$ can be find as
$${\rm im}([f]_{\beta})=\left\{\begin{pmatrix}x\\y\\z\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4615761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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An $n\times n$ matrix $A$ that contains each of $1,2,\cdots, n$ exactly $n$ times and $A^2\equiv 0\pmod n$ Construct a matrix $A \in \{1,\,2, ...,\,n\}^{n \times n}$ such that
(1) Each integer between $[1, n]$ occurs in $A$ exactly $n$ times,
(2) For all integers $i,\,j \in [1, n]$, $(A^2)_{ij} \equiv 0\,(\text{mod}\,n... | We have the following interesting family of examples. If $n$ is divisible by $4$, then take $B$ to be the matrix whose columns are
$$
c_1 = \left(1,3,\dots,\frac n2 - 1, 0\dots,0,\frac n2 + 1,\frac n2 + 3,\dots, n-1\right),\\
c_2 = \left(0,0,\dots,0, 1,\dots,1,0,0,\dots, 0\right),
$$
and take $C$ to be the matrix whose... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4616962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Finding $\cot(\beta)$, knowing $\sin(\alpha+\beta)=4/5$ and $\sin(\alpha-\beta)=3/5$ Let's assume that for $0<\beta<\alpha<\frac{\pi}{2}$, $\sin(\alpha+\beta) = \frac{4}{5}$, and $\sin(\alpha-\beta) = \frac{3}{5}$. Then, how could we find $\cot(\beta)$?
$$\sin(\alpha+\beta)+\sin(\alpha-\beta) = 2\sin(\alpha)\cos(\beta)... | $$
\begin{aligned}
& \sin ^2(\alpha+\beta)+\sin ^2(\alpha-\beta)=\frac{16}{25}+\frac{9}{25}=1 \\
\Rightarrow \quad & \sin ^2(\alpha+\beta)=1-\sin ^2(\alpha-\beta)=\cos ^2(\alpha-\beta)\\ \Rightarrow \quad &
(\sin \alpha \cos \beta+\sin \beta \cos \alpha)^2=(\cos \alpha \cos \beta+\sin \alpha \sin \beta)^2 \\\Rightarro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Maximizing/minimizing $f(\theta) = \sqrt{2}\cos(\theta)-4\sin(\theta)$
Assume that $f : [0, 2\pi]\rightarrow \mathbb{R}$ is a function such that $f(\theta) = \sqrt{2}\cos(\theta)-4\sin(\theta)$. Then, how can we maximize/minimize $f$?
We can re-parametrize our function $f$ by defining another function $g : [-1, 1]\ri... | Use Cauchy-Schwarz inequality: $f^2(\theta) = \left(\sqrt{2}\cos\theta - 4\sin\theta\right)^2\le ((\sqrt{2})^2+4^2)(\cos^2\theta+\sin^2\theta)=18\implies -3\sqrt{2} \le f(\theta) \le 3\sqrt{2}$. Can you conclude the min and max for $f$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
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Evaluate this double Integral in polar coordinates. $$ \iint y^2(a^2-x^2)^{0.5}dxdy $$ over $x^2+y^2\le a^2$
I have evaluated this in $x-y$ plane and got $32a^5/45$. Please help in evaluating the same in polar coordinates.
Ive tried putting $x=r\cos\theta$ and $y=r\sin\theta$ and After considering the Jacobian I am get... | Let $D$ be the disk of radius $a$ centered at the origin. We can parameterize the region in polar coordinates by letting
$$\begin{cases}x = a r \cos(\theta) \\ y = a r \sin(\theta) \end{cases}$$
with $(r,\theta) \in [0,1] \times [0,2\pi]$. The Jacobian in this case is almost identical to the typical one you get when co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4621638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Simplifying $\frac{x^4(y-z)+y^4(z-x)+z^4(x-y)}{(y+z)^2+(z+x)^2+(x+y)^2}$ I need help to simplify this expression.
$$\frac{x^4(y-z)+y^4(z-x)+z^4(x-y)}{(y+z)^2+(z+x)^2+(x+y)^2}$$
I only know to start, I extracted the $(x-y)$ from top one, and then I don't know what's next. Extracting $(x-y)$ from the numerator:
$$(x^4y -... | I like Pavel's answer, but here's a slightly different way.
Starting with $(x-y)(x^3y+x^2y^2+xy^3−zx^3−zy^3+z^4−zxy^2−zx^2y)$, rearrange the longer expression to:
$$(x^3y-zx^3) + (xy^3-zxy^2) + (x^2y^2 - zx^2y) + (z^4 - zy^3)$$
$$ = (y-z)(x^3 + xy^2 + x^2y) - z(z^3-y^3)$$
$$ = (y-z)(x^2y + xy^2 + x^2y -z(y^2+yz+z^2)) $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4622054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show $k \cdot a + c \cdot b \ge \frac{1}{2} \cdot (k+c) \cdot (a+b)$? Assume k, c, a, b $\gt$ 0 and that $k \ge c$ and $a \ge b$. Prove $k \cdot a$ + $c \cdot b$ $\ge$ $\frac{1}{2}\cdot (k+c) \cdot (a+b)$. This is what I have so far:
$2\cdot k\cdot a +2\cdot c \cdot b$ = $(k\cdot a + k\cdot a)$ + $(c\cdot b + c\... | We have $k≥c$ and $a≥b$. And we have to prove that $ka+bc≥(k+c)(a+b)/2$. Now by algebraic expansion we can write it out like these $$ka+bc≥(k+c)(a+b)/2$$ $$or, ka+bc≥kb+ac$$. Since $k≥c$ and $a≥b$ by Appling rearrangement inequality, the result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4622794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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If $a,b>0$ and $a+b=2$ , prove that $a^{2b}+b^{2a}+(\frac{a-b}2)^2\leqslant2$ If $a,b>0$ and $a+b=2$ , prove that
$$
a^{2b}+b^{2a}+(\frac{a-b}2)^2\leqslant2
$$
The equality occurs if and only if $(a,b)\sim(1,1)$ or $(a,b)\sim(2,0)$ or its cyclic permutations.
My attempt:
By symmetry and the constraint, we may let $x=a-... | Variant @Macavity .
We use a form of the Young's inequality wich is a somewhere a generalization of the Bernoulli's inequality :
Let $a,b>0$ and $0<v<1$ then we have :
$$av+b(1-v)\geq a^vb^{1-v}$$
Taking account of this theorem and putting :
$a=(1+x)$$\quad$$b=1$$\quad$$v=(1-x)$ we get $0< x<1$:
$$(1+x)^{2(1-x)}\leq ((... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4624313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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What is the probability f(p) of A winning the match?
Here's my approach:
Let A denote the event "A wins the game" and B denote the event "B wins the game"
I am supposed to calculate the probability f(p)
There are two possible cases:-
i) A wins within 4 games
Possible cases: AAA,ABAA,BAAA,AABA
ii) A doesn't win within ... | I would treat the $2$-$2$ case separately. Calculate the probability that $B$ has won by the end of Game $4$. Then you know the probability of arriving at a $2$-$2$ tie.
$A$ wins within $4$ games with probability $p^3+3p^3(1-p)=4p^3-3p^4=p^3(4-3p)$.
$B$ wins within $4$ games with probability $(1-p)^3+3p(1-p)^3=(1-p)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4627694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Evaluate $(g \circ f)^{-1}(x) = {\left( {\frac{{x - 7}}{2}} \right)^{\frac{1}{3}}}$
If $g\left( x \right) = ax + c$, $f\left( x \right) = {x^b} + 3$, and $(g \circ f)^{-1}(x) = {\left( {\frac{{x - 7}}{2}} \right)^{\frac{1}{3}}}$ what is the value of $a+b+c$?
My approach is as follows
Given $g\left( x \right) = ax + c... | Hint: Recall that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$. You've already compute each inverse separately, so now we just compose them to get:
$$
(g \circ f)^{-1}(x) = f^{-1}(g^{-1}(x))
= f^{-1}\left(\frac{x - c}{a}\right)
= \left( \left(\frac{x - c}{a}\right) - 3 \right)^{\frac{1}{b}}
= \left( \frac{x - 3a - c}{a}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4628817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How do i solve the integral of $x*\ln(x-1)$? I've been having some trouble trying to solve this integral. I feel like i'm going in circles with this. What i did first is use integration by parts in this manner.
$$\int x\cdot \ln(x-1)\;dx = \ln(x-1)\cdot \frac{1}{2}x^2- \int \frac{1}{x-1}\cdot \frac{1}{2}x^2 \; dx$$
$$=... | When you're integrating a rational expression, step one is to make sure it's a proper rational expression. By long division:
$$\frac{x^2}{x-1} = x^2+1+\frac{1}{x-1},$$
which is much easier to integrate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4629013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Unable to crack $\int_0^{\frac1{\sqrt3}} \frac{\cot^{-1}\sqrt{2-x^2}}{1+x^2}dx=\frac{\pi^2}{30}$ I am unable to solve the integral
$$\int_0^{\frac1{\sqrt3}} \frac{\cot^{-1}\sqrt{2-x^2}}{1+x^2}dx=\frac{\pi^2}{30} $$
after a number of attempts, except with a few observations below.
1). Despite the appearance, I do not be... | Perform the substitution $t=\frac{1}{\sqrt{2-x^2}}$ to check that,
$$
\tag{1}\int_{0}^{\frac{1}{\sqrt{3} } } \frac{\operatorname{arccot} \left ( \sqrt{2-x^2} \right ) }{
1+x^2}\text{d}x=
\int_{\frac{\sqrt{2}}{2} }^{\sqrt{\frac{3}{5}}}
\frac{\arctan\left ( t \right ) }{\sqrt{2t^2-1}(3t^2-1) }\text{d}t.
$$
The second in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4630584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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How to solve the given differential equation?
Given $T$ is a function in $t$ $(x>0)$. And $T(@t=0)=40$. And which satisfies,
$$ \frac{dt}{4}= \frac{3\cdot dT}{2} + \frac{T\cdot dt}{400+t}$$
Then find $T$
I tried putting $400+t=u$ and then,
$$ \frac{du}{4}= \frac{3\cdot dT}{2} + \frac{T\cdot du}{u}$$
And multiplying b... | Under your substitution, the ODE becomes homogeneous
$$\frac32 \, dT = \left(\frac14 - \frac T{400+t}\right) \, dt \implies \frac{dT}{du} = \frac16 \left(1-\frac{4T}u\right)$$
Further substituting $U=\frac Tu$ and $\frac{dU}{du}=\frac{u\frac{dT}{du} - T}{u^2}$ gives the separable ODE,
$$u\,\frac{dU}{du} + U = \frac{1-4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4635208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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For odd $k$, $\sum_{j=0}^{N-1}\cos^k(\pi j/N) = 1$. Is there a nicer proof? Is this well known? In the course of answering this question, I stumbled into the following result: for any integer $N > 1$ and any odd $k$,
$$
\sum_{j=0}^{N-1}\cos^k(\pi j/N) = 1.
$$
My proof, which I suspect is much more complicated than it n... | The result follows since $\cos \pi j/N=-\cos \pi (N-j)/N$ for $j=1,..N-1$, while for $j=0$ we get $1$. Now $a=-b$ implies $a^k=-b^k$ for any odd $k$. So all but the first term cancel in pairs (and the middle term is zero for $N$ even)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4635961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Number of possibilities for creating words of length four A random generator is generating words of length $4$ out of the alphabet $\{0,1,2,3,4,5,6,7,8,9\}$, for example $1234$.
What is the probability to get
*
*all different numbers?
*exactly one pair of same numbers?
*exactly two pairs of the same numbers?
As t... | The first answer is correct. The second and third are incorrect since it does not matter which of the values is selected first when you pick two numbers that appear the same number of times.
Exactly one pair of same numbers: There are $10$ ways to select the number which appears twice, $\binom{4}{2}$ ways to select t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the $n$th power of a 3-by-3 circulant matrix Consider the matrix given as $$A=\begin{bmatrix}a_0 & a_2 & a_1\\ a_1 & a_0 & a_2\\ a_2 & a_1 & a_0\end{bmatrix}$$
Write down a formual for $A^n$ for $n\in\mathbb{N}$.
$$$$
My attempt: The first that comes to mind is to diagonalize it and hence find the formual for $A^n... | I think you've reached a decent result and by just using the multinomial theorem you can obtain a generalized solution.
$A=a_0I+ a_1U + a_2 U^2 \\
A^n = (a_0I+ a_1U + a_2 U^2)^n\\$
Then the coeffecients of $U^3$, $U^2$ and $U^1$ are determined using the multinomial theorem and just plugged back into the result matrix.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4639166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Prove that $\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n$ Let $n$ be a positive integer and $x_1,...,x_n$ positive reals such that $x_1+...+x_n=1$. Prove that $$\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n$$
Now notice that if we apply Jensen's Inequality to $$f(x)=\ln\left(\frac{1}{x_i^2}-1\rig... | Lemma:$$\left(\dfrac{1}{x}-x\right)\left(\dfrac{1}{y}-y\right)\ge 4^3\left(\dfrac{z^3}{3^9\cdot x^2y^2}\right)^{\frac{1}{4}}~~~~~(n=3)$$
$$
LHS=\dfrac{1}{xy}\left(\dfrac{z}{3}+\dfrac{z}{3}+\dfrac{z}{3}+xy\right)\left(x+\dfrac{1}{3}+\dfrac{1}{3}+\dfrac{1}{3}\right)\left(y+\dfrac{1}{3}+\dfrac{1}{3}+\dfrac{1}{3}\right)\ge... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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$(a+1)x^{2} +(b+1)y^{2} +(c+1)z^{2} \geq 2(xy+yz+zx)$ Prove $(a+1)x^{2} +(b+1)y^{2} +(c+1)z^{2} \geq 2(xy+yz+zx)$ for $a,b,c>0$ and $abc\geq1$.
My try: $(a+1)x^2-4xy+(b+1)y^2+(a+1)x^2-4xz+(c+1)z^2+(b+1)y^2-4yz+(c+1)z^2\geq0$
Then I had to prove that $(a+1)x^2-4xy+(b+1)y^2\geq2x^2-4xy+2y^2$, but I realized that there's ... | We can assume at least two of ${x}$, ${y}$, ${z}$ non negative, otherwise we take their additive inverse. Let ${y}$, $z\ge0$.
Consider $x{}$ the only variable. The inequality can be regarded as a quadratic function of $x$. Since the coefficient of $x^2$is positive, we only need $\frac\Delta4\le0$, which requires
\[(a+1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the maximum value of $a^2(b-c)+b^2(c-b)+c^2(1-c)$, if $0\le{a}\le{b}\le{c}\le{1}$.
How do I find the maximum value of $a^2(b-c)+b^2(c-b)+c^2(1-c)$, if $0\le{a}\le{b}\le{c}\le{1}$?
I have tried plugging in values, tried manipulating and inequality by means but that didn't worked. How do I approach this question?
| Capitalizing on writing the expression as a sum of two positive numbers, indicates the value of a:
$$\begin{align}E(a,b,c)&=\underbrace{(b^2-a^2)}_{\ge 0}\underbrace{(c-b)}_{\ge 0}+\underbrace{c^2(1-c)}_{\ge 0}, a,b,c \lt 1\\ &\le (b^2-0)(c-b)+c^2(1-c), \boxed{a= 0}\end{align}$$
The first term $b^2(c-b)$ could be analy... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Fraction of prime number between consecutive powers of two A $n$-bit integer is an integer $x$ such that $2^{n-1} \le x < 2^n$. In [1] it is claimed without proof that a corollary of the prime number theorem is that:
For any $n > 1$, the fraction of $n$-bit integers that are prime is at least $\frac{1}{3n}$.
Can some... | This is just a more detailed write-up of the strategy proposed by Marty Cohen's answer.
We can prove the stronger claim: for $n \ge 1$ there are at least $\frac{2^{n-2}}{2n}$ $n$-bit prime numbers (i.e., the fraction of $n$-bit prime numbers is at least $\frac{1}{2n}$).
According to [1], for $x \ge 55$,
$\dfrac{x}{\log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4643160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to evaluate $\int x \sqrt{x^2 - x}\ dx$
Problem is to integrate $ x \sqrt{x^2 - x}$.
My attempt:
I made it ready for a substitution $u = x^2 - x$
$$\begin{aligned} \int x \sqrt{x^2 -x}\ dx &= \int (2x-1)\sqrt{x^2 - x} \ dx - \int(x - 1)\sqrt{x^2 - x}\ dx\\& = \int \sqrt u\ du - \int(x - 1)\sqrt{x^2 - x}\ dx\\& ... | Hyperbolic function for $\int \sqrt{u^2-u}du$, using integration by parts yields
$$
\begin{aligned}
K & =u \sqrt{u^2-1}-\int \frac{u^2-1+1}{\sqrt{u^2-1}} d u \\
& =u \sqrt{u^2-1}-K-\int \frac{d u}{\sqrt{u^2-1}}
\end{aligned}
$$
Putting $y=\cosh u$ yields
$$
K=\frac{1}{2}\left(u \sqrt{u^2-1}-\operatorname{arccosh} u\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4646982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 4
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Combinatorics: Amount of options for majority Say we have $2n$ people. Then the amount of options to form a majority (e.g. in a commission) are
$ \binom{2n}{n+1} + \binom{2n}{n+2} + \cdots + \binom{2n}{2n}$
I want to prove, that this is equal to $\frac{1}{2} \left[ 2^{2n} - \binom{2n}{n} \right]$
But I'm stuck. I have ... | By the property of binomial coefficients $$\binom{n}{r}=\binom{n}{n-r}$$
$\implies \binom{2n}{n+1}=\binom{2n}{n-1}$
$\implies\binom{2n}{n+2}=\binom{2n}{n-2}$ and so on
Let $$A=\binom{2n}{n+1}+\cdots+\binom{2n}{2n}$$ Also
$$A=\binom{2n}{0}+\cdots+\binom{2n}{n-1}$$
So $$2A=2^{2n}-\binom{2n}{n}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4647482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving Generating Function holds a specific recurrence. Consider the generating function $$\dfrac{1}{1 − 2x − x^2} = \sum_{n=0}^{\infty}a_nx^n$$
Prove that for each integer $n \ge 0$,
$$a_n^2+a_{n+1}^2 = a_{2n+2}$$
Hint: Find a $2 \times 2$ matrix $A$ such that
$$A^{n+2} =\begin{bmatrix}a_n &a_{n+1}\\
a_{n+1} &a_{n+2}... | Define $f(x)$ by
$$f(x)= \frac{1}{1-2x-x^2} = \sum_{n=0}^{\infty} a_n x^n$$
so
$$f(x) -2x f(x) -x^2 f(x) = 1$$
From this equation we deduce $a_0=1$, $a_1=2$, and $a_{n+2} -2a_{n+1} -a_n = 0$ for $n \ge 0$. It follows by induction on $n$ that if we let
$$A = \begin{pmatrix}
0 &1 \\
1 &2 \end{pmatrix}$$
then
$$A^{n+2} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4648320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the exact value of $\int_{0}^{2}x[\frac{1}{x}]dx$. Find the exact value of $\int_{0}^{2}x[\frac{1}{x}]dx$.
Let $[x]$ denote $\lceil{x-\frac{1}{2}}\rceil$.
Using Desmos, I got $2.46736022133$ and WolframAlpha does not give me a solution. My intuition tells me that it might be possible to find an exact value using T... | $$\frac{16 n^2}{\left(4 n^2-1\right)^2}=-\frac{1}{2 n+1}+\frac{1}{(2 n+1)^2}+\frac{1}{2 n-1}+\frac{1}{(2 n-1)^2}$$
$$\sum_{n=1}^p\frac{16 n^2}{\left(4 n^2-1\right)^2}=\frac{1}{2} \left(\psi ^{(0)}\left(\frac{3}{2}\right)-\psi
^{(0)}\left(p+\frac{3}{2}\right)\right)+$$
$$\frac{1}{8} \left(-2 \psi ^{(1)}\left(p+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4650684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Proving that $0 \leq k \sin\left(\frac{2\pi}{n}\right) - \sin\left(\frac{2\pi k}{n}\right)$ I need help proving that
$$
0 \leq k \sin\left(\frac{2\pi}{n}\right) - \sin\left(\frac{2\pi k}{n}\right)
$$
For $n > 2$ and positive $k$.
I've tried all sorts of identities, and nothing have worked. Any help would be greatly app... | I am assuming that $k,n\in\mathbb{Z}$.
Suppose that for some $k$, we have
$$
\sin\left(\frac{2\pi k}{n}\right)\le k\sin\left(\frac{2\pi}{n}\right)\tag1
$$
Note that $(1)$ is trivially true for $k=1$.
For $k\ge2$, we have
$$
k\,\overbrace{\sin\left(\frac{2\pi}4\right)}^1\gt k\,\overbrace{\sin\left(\frac{2\pi}3\right)}^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
| $$\sum_{i=0}^ni-\sum_{i=0}^{n-1}i=S_1(n)-S_1(n-1)=n,$$
so that $S_1(n)$ must be a polynomial of the second degree in $n$.
By the method of undeterminate coefficients, noting that there is no constant term as $S_1(0)=0$:
$$S_1(n)-S_1(n-1)=n=(an^2+bn)-(a(n-1)^2+b(n-1))=\\=2an+b-a,$$
and by identification
$$a=b=\frac12.$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "136",
"answer_count": 36,
"answer_id": 17
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Explain why calculating this series could cause paradox? $$\ln2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - 2(\frac{1}{2} + \frac{1}{4} + \cdots)$$
$$= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - (1 + \frac{1}{2} + \frac{1}{3}... | In Matt's answer I began discussing with Matt E over several comments, which I think should be written out as an answer.
As Matt pointed out, this is a rearrangement of this conditionally convergent series which is why you have this sort of paradox.
However it was unclear about how this is exactly a rearrangement, as t... | {
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"source": "stackexchange",
"question_score": "5",
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How to prove an identity in radicals? (4 / (3 - sqrt(5))) ^ 2 - ((6 - 5 * sqrt(6)) / (5 - sqrt(6))) ^ 2 = 2 * sqrt(61 + 24*sqrt(5))
$$\left(\frac{4}{3-\sqrt{5}}\right)^2 - \left(\frac{6 - 5 \sqrt{6}}{5 - \sqrt{6}}\right)^2 = 2\sqrt{61+24\sqrt{5}}$$
How to prove it is right equality?
I come up with $\dfrac{16}{14-6\sqrt... | Trivial. Lets simplify the left part:
multiply first fraction to the sum of expressions in its denominator, and get whole part in second fraction and do with second fraction the same that we did with first:
$(\frac{4(3+\sqrt{5})}{9-5})^2 - (\frac{25-19 - \sqrt{6}}{5-\sqrt{6}}) = (3+\sqrt{5})^2 - (5-\frac{19(5+\sqrt{6})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/5937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Partial sum of the series $\sum\limits_{r=1}^{\infty} \frac{1}{r(r+1)}$ An Exercise from Apostol's Introduction to Analytic Number Theory which I am not able to solve.
Let $\mathsf{S_{n}}$ denote the $n$-th partial sum of the series: $$\sum\limits_{r=1}^{\infty} \frac{1}{r(r+1)}$$
Prove that for every integer $k>1$, th... | (EDIT: My original post took $p$ to be the smallest prime that divides $k$. This is unnecessary. You can take $p$ to be any number other than 1 that divides $k$. Doing so gives you not just one solution but $d(k)-1$ solutions, one for each of the divisors of $k$ other than 1.)
(SECOND EDIT: See the answer to this ... | {
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"source": "stackexchange",
"question_score": "4",
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$\sum\limits_{n=0}^{\infty} \frac{1}{(2n+1)!}$ converges? Determine whether this series converges or diverges: $$\sum\limits_{n=0}^{\infty} \frac{1}{(2n+1)!}$$
Thought about using the limit theorem or by comparison but am so stuck. any pointers would be appreciated guys
| Another way is
If $\displaystyle S_n = 1 + \frac{1}{3!} + \dots + \frac{1}{(2n+1)!}$
We have that
$\displaystyle S_n \le 1 + \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n(n+1)}$
$\displaystyle = 1 + (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + \dots + (\frac{1}{n} - \frac{1}{n+1}) = 2 - \frac{1}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 5
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An inequality like Riemann sum involving $\sqrt{1-x^2}$ How can I prove that for every positive integer $n$ we have
\begin{equation*}
\frac{n\pi}{4}−\frac{1}{\sqrt{8n}}<\frac{1}{2}+\sum_{k=1}^{n−1}\sqrt{1−\frac{k^2}{n^2}}?
\end{equation*}
| Write the inequality as
$$\frac{\pi}{4} < \frac{1}{2n} + \frac{1}{n} \sum_{k=1}^{n-1} \sqrt{1-\left(\frac{k}{n}\right)^2} + \frac{1}{2n} \sqrt{\frac{1}{2n}}.$$
The left-hand side $\pi/4$ is the area of the part of the unit circle that
lies in the first quadrant (below the curve $y=f(x)=\sqrt{1-x^2}$).
We want to interp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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The Basel problem As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it w... | Here's mine. I'm answering late, I know that, but I am still answering it.
We'll use the expansion of $\tanh^{-1}$:
$$\frac{1}{2}\log\frac{1+y}{1-y}=\sum_{n\geq0}\frac{y^{2n+1}}{2n+1},\quad|y|<1$$
We start with this inequality:
$$\int_{-1}^{1}\int_{-1}^{1}\frac{1}{1+2xy+y^2}dy\,dx=\int_{-1}^{1}\frac{1}{1+2xy+y^2}dx\,dy... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "814",
"answer_count": 48,
"answer_id": 28
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Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$ How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
| Here's a proof that only requires elementary, but clever, calculus manipulations.
Let $I_A = \int_0^A e^{-x^2}dx$.
$$\begin{split} I_A^2 &= \int_0^A \int_0^A e^{-(x^2+y^2)}dxdy\\
&= \int_0^A \int_0^A \sum_{n\geq 0} \frac{(-1)^n}{n!}(x^2+y^2)^ndxdy\\
&=\sum_{n\geq 0} \frac{(-1)^n}{n!}\int_0^A \int_0^A\sum_{k=0}^n{n \cho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "210",
"answer_count": 18,
"answer_id": 10
} |
Prove by induction $T(n) = 2T(\frac{n}{2}) + 2$ I'm stuck with this induction proof:
So far, given:
$\begin{align*}
T(1) & = 2 \\
T(n) & = 2T(n/2)+2 \\
& = 2(2T(n/[2^2])+2) + 2 \\
& = [2^2]T(n/[2^2]) + [2^2] + 2 \\
& = [2^2](2T(n/[2^2])+2) + [2^2] + 2 \\
& = [2^3]T(n/[2^3]) + [2^3] + [2^2] + 2 \\
& = [2^3]T(n/[2^3]) + ... | Here is how I attempt this from the point you left off:
We know that $\rm T(1) = 2$. We are trying to prove that $\rm T(n) = 4n-2$.
This is trivially true for $n = 1$:
$ \rm \begin{eqnarray*}
T(1) &=& 4(1) - 2\\
&=& 4 - 2\\
&=& 2 \\
\end{eqnarray*}
$
Assume
$\rm T(k) = 4k - 2 $
From the original def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Inequality of three variable polynomial I read that one can prove by AM-GM-inequality that for all $a,b,c\in\mathbb{R}_+$ we have that
$$11(a^6 + b^6 + c^6) + 40abc(ab^2 + bc^2 + ca^2) \ge 51abc(a^2b + b^2c + c^2a)$$
How this can be done? Is it possible to prove stronger inequalities like
$$10(a^6 + b^6 + c^6) + 41abc... | You want to apply AM-GM to some of the terms on the LHS to get some of the terms on the RHS. A naive way to do this such as
$$b^6 + c^6 + 2a^2 b^3 c \ge 4 a b^3 c^2$$
uses too many of the $11$ terms relative to the $40$ terms and only proves the weaker inequality
$$a^6 + b^6 + c^6 + abc(ab^2 + bc^2 + ca^2) \ge 2abc(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/11189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
An inequality on Cevians Let $\displaystyle AD$, $\displaystyle BE$, $\displaystyle CF$ be three cevians concurrent at $\displaystyle P$ inside the $\displaystyle \Delta ABC$.
Prove or disprove that:
$$\displaystyle \dfrac{AD}{AP} + \dfrac{BE}{BP} + \dfrac{CF}{CP} \ge \dfrac{9}{2}$$
| The inequality is true!
It can be shown that (see proof at the end of the answer)
$$\displaystyle \frac{PD}{AD} + \frac{PE}{BE} + \frac{PF}{CF} = 1$$
Note that this implies that
$$\displaystyle \frac{AP}{AD} + \frac{BP}{BE} + \frac{CP}{CF} = 2$$
as $\displaystyle 1 - \frac{PD}{AD} = \frac{AP}{AD}$ etc.
Now we have th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/12929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
How can I find the derivative of $y = \sin(\arctan x) + \tan(\arcsin x)$? My question is, how can I solve the following derivative question?
$$y = \sin(\arctan x) + \tan(\arcsin x)$$
Thanks in advance
| First you want to simplify $\tan (\arcsin x)$ and $\sin(\arctan x)$. Let me show
you the latter one.
Suppose $\theta=\arctan x$. Then by definition $-\pi/2<\theta<\pi/2$ and $\tan\theta=x$.
Since $\sec \theta>0$ (because $-\pi/2<\theta<\pi/2$), and $$\sec^2\theta=1+\tan^2 \theta =1+x^2,$$ you get
$\sec \theta=\sqrt{1+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/15514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Algebra Problem The expression $x^2-4x+5$ is a factor of $ax^3+bx^2+25$. Express the sum a+b as an integer.
Please give an explanation of how the answer
| $\rm\ 0 = a\ x^3 + b\ x^2 + 25 - (x^2 - 4\ x + 5)\ (a\ x + 5) = (4\ a + b - 5)\ x^2 + (20 - 5\ a)\ x\ \Rightarrow \ a,\: b = \ldots$
Note: $\rm\ a\ x + 5\ $ comes from comparing leading and constant coefficients.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/16284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Solve $\cos(\theta) + \sin(\theta) = x$ for known $x$, unknown $\theta$? After looking at the list of trigonometric identities, I can't seem to find a way to solve this. Is it solvable?
$$\cos(\theta) + \sin(\theta) = x.$$
What if I added another equation to the problem:
$$-\sin(\theta) + \cos(\theta) = y,$$
where $\th... | Another method goes by noting that $\cos^2\theta+\sin^2\theta=1$. We have $\cos\theta+\sin\theta=x$, so $$\cos^2\theta+2\cos\theta\sin\theta+\sin^2\theta=x^2,$$ or $2\cos\theta\sin\theta=x^2-1$. But $2\cos\theta\sin\theta=\sin(2\theta)$, so $2\theta=\sin^{-1}(x^2-1)$, or $$ \theta=\frac12\sin^{-1}(x^2-1).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/20548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 0
} |
Evaluate $\int \cos^3 x\;\sin^2 xdx$ Is this correct? I thought it would be but when I entered it into wolfram alpha, I got a different answer.
$$\int (\cos^3x)(\sin^2x)dx = \int(\cos x)(\cos^2x)(\sin^2x)dx
= \int (\cos x)(1-\sin^2x)(\sin^2x)dx.$$
let $u = \sin x$, $du = \cos xdx$
$$\int(1-u^2)u^2du = \int(u^2-u^4... | \begin{align*}
\frac{\sin^3 (x)}{3} - \frac{\sin^5 (x)}{5} &= \sin^3 (x) (\frac{1}{3} - \frac{\sin^2 (x)}{5});\\
\cos(2x) &= 1 - 2 \sin^2(x)\\
\sin^2(x) &= \frac{1- \cos(2x)}{2}.
\end{align*}
Hence, we get,
\begin{align*}
\sin^3 (x) \left(\frac{1}{3} - \frac{\sin^2 (x)}{5}\right) &= \sin^3 (x) \left(\frac{1}{3} - \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Find lowest integer value with integer x of given function I need to find integer x with which function's y gets lowest integer values
$$f(x)=\frac{x^2-x-17}{x-2}$$
I tried to find derivative, but it never equals 0.
Other steps was to change expression
$$f(x)=\frac{x^2-x-17}{x-2}=1+\frac{(x-5)(x+3)}{x-2}=x-2+\frac{3(x-... | For $n < -13$ we have $n+1 < f(n) < n+2$ and so $f(n)$ is not an integer for integer $n$.
For $n > 17$ we have $n < f(n) < n+1$ amd so $f(n)$ is not an integer for integer $n$.
So you just have to check the 31 integers remaining. The integer values are $f(-13)=-11$, $f(-3)=1$, $f(-1)=5$, $f(1)=17$, $f(3)=-11$, $f(5)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/25780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$?"
Are there any nice ways to prove that $$\z... | Consider the contour integral
$$
\oint_C\frac{\pi\cot\pi z}{z^4}\ dz
$$
where $C$ is the counter-clockwise square contour centered at origin with vertices $\left(N+\frac12\right)(\pm1\pm i)$.
Lemma
Suppose that the function $\phi(z)$ is holomorphic at $z=n\in\mathbb{Z}$ with $\phi(n)\neq0$, then $\pi\phi(z)\cot\pi z$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/28329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "119",
"answer_count": 15,
"answer_id": 10
} |
Can this integral $\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$ be written in the form of a elliptic integral I am trying to find the magnetic field due to an elliptic loop of wire. How to do integrals of the type $$\int_0^{2\pi} \frac{d\theta}{(a^2 \cos^2 \theta +b^2\sin^2\theta)^{3/2}}$$... | (I assume that $a > b$ in this answer.)
Hans's answer covers the OP's needs, but I'll try out the pedestrian approach here.
First, a trivial simplification:
$$\int_0^{2\pi} \frac{\mathrm d\theta}{(a^2 \cos^2\theta +b^2\sin^2\theta)^{3/2}}=\int_0^\pi \frac{\mathrm d\theta}{(a^2 \cos^2\theta +b^2\sin^2\theta)^{3/2}}+\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/34516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 0
} |
More Computing Integrals This particular problem has been giving me trouble, and while the math dept tutors did help a great deal, the resulting answer hasn't been accepted by the online homework submission website. Find the definite integral of
$$\int\frac{x(x+1)}{2x^{3}+3x^{2}-13}$$
The work done so far with the hel... | What's missing is what you're integrating with respect to. You need to put a $ dx $ in the first 3 integrals of your post (and a $ du $ in any integrals involving $ u $).
Our integral is $ \int \frac{x^2 + x}{2x^3 + 3x^2 - 13} \ dx $. We make the substitution $ u = 2x^3 + 3x^2 - 13 $. This gives us $ \frac{du}{dx} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/35735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
A differential equation (nonlinear First-Order) how to solve this equation:
$(Px-y)(Py+x)=h^2P$
that $P=\frac{dy}{dx}$
and $h$ is a constant.
| Assume $h\neq0$ for the key case:
Let $u=x^2+y^2$ ,
Then $\dfrac{du}{dx}=2x+2y\dfrac{dy}{dx}$
$\therefore\left(\dfrac{x}{2y}\left(\dfrac{du}{dx}-2x\right)-y\right)\dfrac{1}{2}\dfrac{du}{dx}=\dfrac{h^2}{2y}\left(\dfrac{du}{dx}-2x\right)$
$\left(x\left(\dfrac{du}{dx}-2x\right)-2y^2\right)\dfrac{du}{dx}=2h^2\left(\dfrac{d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/36625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Recurrence relation (linear, second-order, constant coefficients) Q1. Find the general solution to the difference equation
$$ a_{n} - 4a_{n-1} + 3a_{n-2} = 6 $$
Q2. Solve the difference equation
$$ a_{n} - a_{n-1} - 2a_{n-2} = 0, a_0 = 2, a_1 = 4 $$
I am completely lost in solving recurrence relation questions. Ca... | Simple, general way of disposing such equations is using generating functions.
Define:
$$
A(z) = \sum_{n \ge 0} a_n z^n
$$
Write the recurrence so it hasn't subtractions in indices:
$$
a_{n + 2} - 4 a_{n + 1} + 3 a_n = 6
$$
Multiply by $z^n$, sum over $n \ge 0$ an recognize the resulting terms:
$$
\frac{A(z) - a_0 - a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/37157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Where are these additional solutions coming from? Solve for $x$: $2\sin(2x)-\sqrt{2} = 0$ in interval $[0,2\pi)$
Step $1$: Add $\sqrt{2}$ and divide by $2$ to get $\sin(2x) = \dfrac{\sqrt{2}}{2}$
Step $2$: Set $2x$ equal to the angles where $\sin(x) = \dfrac{\sqrt{2}}{2}$:
$2x = \dfrac{\pi}{4}$ and $2x = \dfrac{3\pi... | $x\in[0,2\pi)$ means $2x\in[0,4\pi)$, so you have find the solutions in this larger interval.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/38583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Solving $\sqrt{x+5} = x - 1$ I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it:
$$\sqrt{x+5} = x - 1$$
So I used this logic:
$$
\begin{align}
\sqrt{x+5} &= x - 1 \\
x + 5 &= (x-1)^2 \\
x + 5 &= (x-1)(x-1) \\
x + 5 &= x^2 - 2x + 1 \\
0 ... | The problem is:
$$x+5 = (x - 1)^2$$
doesn't imply
$$\sqrt{x + 5} = x - 1$$
but
$$\sqrt{x + 5} = \vert x - 1\vert$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/41152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 7,
"answer_id": 4
} |
Confused by textbook solution to trig problem The following is part of a question and the solution from my textbook:
Question
Given that $2 \sin{2\theta} = \cos{2\theta}$, show that $\tan{2\theta} = 0.5$
Solution
$2 \sin{2\theta} = \cos{2\theta}$
$\Rightarrow \frac{2 \sin{2\theta}}{\cos{2\theta}} = 1$
$\Rightarrow 2 ... | If $\cos(2\theta) = 0$, then solving for $\theta$ we get one value of $\theta =\frac{\pi}{4}$, which says that the value of $$2 \sin{2\theta} = 2 \sin\frac{\pi}{2} = 2 \neq 0$$
which doesn't satisfy the given hypothesis $2 \sin{2\theta} = \cos{2 \theta}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/41222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Limit of eigenvectors versus eigenvectors of limit Consider the matrix
$$
A=\begin{pmatrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & g
\end{pmatrix},
$$
where $g$ is a real parameter. If I set $g=0$ and calculate the normalized eigenvectors of $A|_{g=0}$ with Mathematica, I find that they are
$$
v_1 = \frac{1}{\sqrt... | Both $(-1,1,0)^T$ and $(-1,0,1)^T$ are eigenvectors of $-1$; so is $(-1,-1,2)^T$, as
$$(-1,-1,2) = -(-1,1,0) + 2(-1,0,1).$$
You are just taking a different basis for the eigenspace corresponding to $-1$.
It's likely just an artifact of how Mathematica finds a basis for the eigenspace; the eigenvalues of the matrix are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Transforming power series Definition:
Let $$a_0 = a_1 = 1, \; a_{n+2} = a_{n+1} + (n+1) \cdot a_n, \; n \geq 0$$
Exercise:
Prove that $$\sum_{n\geq 0} \frac{a_n}{n!} x^n = \exp \left( x + \frac{1}{2} x^2 \right)$$
I don't know how to start with this. I do know that $$e = \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^... | Let $$S(x)= \sum_{n\ge 0} \frac{a^n}{n!}x^n$$
and note that
$$S'(x)= \sum_{n\ge 1} \frac{a^n}{(n-1)!}x^{n-1}$$
Replacing in the first equation the value of $a_n$ given by the recursion, $a_{n}=(a_{n+2}-a_{n+1})/(n+1)$ we get
$$S(x) = S_2(x) - S_1(x)$$
where
$$S_1(x) = \sum_{n\ge 0} \frac{a^{n+1}}{(n+1)!}x^n =\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How to find remainder modulo $n$, when $n$ is a large number I am doing RSA questions and I really could use help! Can someone show me a simple way to find $25^9 \pmod{33}$?
| In the case of $25^9 \pmod{33}$ it is pretty straightforward to use exponentiation by squaring because $9 = 8 + 1 = 2^3 + 2^0$.
$25 \equiv 25 \pmod{33}$
$25^2 \equiv 31 \pmod{33}$
$25^4 \equiv 31^2 \equiv 4 \pmod{33}$
$25^8 \equiv 4^2 \equiv 16 \pmod{33}$
$25^9 \equiv 25^8 \times 25 \equiv 16 \times 25 \equiv 4 \pmod{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 5
} |
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 ... | Firstly, calculate the sum $S=\sum_{k=1}^n k(k+1)$ which is:
$S=1\cdot 2+2\cdot 3+ \cdots +n(n+1)$,
multiplying $S$ by 3
we get:
$3S=1\cdot 2\cdot 3+2\cdot 3\cdot (4-1)+3\cdot 4\cdot (5-2)+ \cdots +n\cdot (n+1)\cdot (n+2-(n-1))$
$3S=1 · 2 · 3 + 2 · 3 · 4 − 1 · 2 · 3 + 3 · 4 · 5 − 2 · 3 · 4 + · · ·
+ n(n + 1)(n + 2) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/48080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "145",
"answer_count": 32,
"answer_id": 27
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Inequality: $(x + y + z)^3 \geq 27 xyz$ Edit: $a,b,c$ and $x,y,z$ are positive, real numbers.
Since $(a-b)^2 \geq 0~$, $a^2 + b^2 - 2ab\geq0~$ and $a^2 + b^2 \geq 2ab~$. Similarly, $a^2 + c^2 \geq 2ac~$ and $b^2 + c^2 \geq 2bc~$.
Adding these inequalities together, $2(a^2 +b^2 + c^2) \geq 2(ab + ac +bc)~$ and accord... | My favorite technique for proving symmetric inequalities of positive numbers (particularly if you have a computer algebra package) is to note that if the inequality is symmetric, then w.l.o.g. we can assume the variables are in sorted order, then rewrite the inequality using the smallest variable and the consecutive di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/48621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 8,
"answer_id": 3
} |
Logarithm rules What can I do with these expression: $2^{\log _{\frac{4}{3}}n}$ and $2^{\log _{4}n}$ if I don't want to have $n$ in the exponent?
I tried nothing because I didn't have any good ideas.
Thanks.
| For example, $2^{\log_4n}=2^{2\cdot\frac{1}{2}\cdot\log_4n}=((2^{2})^{\log_4n})^{\frac{1}{2}}=(4^{\log_4n})^{\frac{1}{2}}=n^{\frac{1}{2}}=\sqrt{n}.$ In this case, I am using the fact that $2^2=4$ and the property that says that $a^{\log_ab}=b.$
For the other expression, we would need to use that $2^{\log_2{\frac{4}{3}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/49600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Number of Ways of Writing $n$ as Sum I have a general question about numbers:
How many ways can we write $n$ as the sum of the numbers $1$,$2$ and $3$?
I know that we start with the following functions: $$\frac{1}{1-z} = 1+z+z^2+ \dots$$ $$\frac{1}{1-z^2} = 1+z^2+ z^4+ z^6 + \dots$$ $$\frac{1}{1-z^3} = 1+ z^3+ z^{6}+... | Denote by $N_3=\{1,2,3\}\,$ then your question can be reformulated:
Find the number of partitions of natural number n in parts over set $N_3$ that number we denote
$p(n,N_3)$, $n=6k+l$, where $ k\in \{0,1,2,...\}$,and $ l\in \{0,1,2,3,4,5\}\,$
Using recurrences (no generating functions) I find that exact solution is.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/51630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Get number of elements of a square matrix given a vector that has upper right elements of that matrix Suppose I have some vectors:
*
*$[1,2,1]$ its length is $3$ wich represents a matrix like $\begin{pmatrix}
1 & 2\\
-1 & 1
\end{pmatrix}$ the complete matrix would have $4$ elements
*$[1,2,1,3,4,1]$ its len... | As Billy explains, you can find an explicit formula by solving a quadratic, the result of which appears in Fabian's answer.
An alternative is to note that $$\frac{m^2}{2}\lt\frac{m(m+1)}{2}\lt\frac{(m+1)^2}{2}.$$ You are given $n=\frac{m(m+1)}{2}$, and you want $m^2$. The inequalities above yield $$m^2<2n<(m+1)^2,$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Integrate square of the log-sine integral: $\int_0^{\frac{\pi}{2}}\ln^{2}(\sin(x))dx$ $\displaystyle \int_{0}^{\frac{\pi}{2}} \ln(\sin(x))dx=-\frac{\pi}{2}\ln(2)$ is an integral that is common.
But, how can we show $\displaystyle\int_{0}^{\frac{\pi}{2}}\ln^{2}(\sin(x))dx=\frac{{\pi}^{3}}{24}+\frac{\pi}{2}\ln^{2}(2)$?.
... | Alternative solution:
\begin{align}
A&=\int_0^{\frac{\pi}{2}}x\ln^2(\sin x)dx\\
&\overset{y=\sin x}=\int_0^1 \frac{\ln^2 y\arcsin y}{\sqrt{1-y^2}}dy\\
&\overset{\text{IBP}}=\left[\left(\int_0^y \frac{\ln^2 t}{\sqrt{1-t^2}}dt\right)\arcsin y\right]_0^1-\int_0^1 \frac{1}{\sqrt{1-y^2}}\left(\int_0^y \frac{\ln^2 t}{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/58654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 6,
"answer_id": 4
} |
Solve by induction: $n!>(n/e)^n$ To Prove : $n! > (n/e)^n$
The question seems easy but it ain't; anyone up for it ?
| I had originally written this up for another question but it seems fitting here as well. Maybe this can help someone.
Depending on how you introduced $e$, you might be able to use the fact that there are two sequences $(a_n)_{n \in \mathbb{N}}$, $(b_n)_{n \in \mathbb{N}}$ with
$$\begin{align}
a_n ~~~&:=~~~ \left ( 1 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/58711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 1
} |
Determinant of a special skew-symmetric matrix Simple calculation show that:
$$
\begin{align}
\det(A_2)=\begin{vmatrix}
0& 1 \\
-1& 0
\end{vmatrix}&=1\\
\det(A_4)=\begin{vmatrix}
0& 1 &1 &1 \\
-1& 0 &1&1\\
-1& -1& 0&1\\
-1& -1& -1&0
\end{vmatrix}&=1
\end{align}
$$
Here is my questio... | Let $A_{2n}$ the $2n\times 2n$ matrix defined by $(A_{2n})_{i,j} = \begin{cases}
1&\mbox{if }i<j\\
0&\mbox{if }i=j\\
-1&\mbox{if } i>j
\end{cases}$.
We can show the result ($\det(A_{2n}) = 1$) by induction. The case $n=1$ and $n=2$ has been solved. Now, we can write $A_{2n+2}=\begin{pmatrix}A_{2n}&B\\-B^t &C\end{pmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/58935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 4
} |
How many distinct real roots does $ (x^2 + x – 2)^3 + (8–2x^2 )^3 = (x^2 + 3x + 2)^3 $have?
How many distinct real roots does $ (x^2 + x – 2)^3 + (8–2x^2 )^3 =
(x^2 + 3x + 2)^3 $have?
If I didn't make any mistake that equation could be reduced to the form:$$(2+6x^2)(x+2)^3=(8–2x^2 )^3$$
However,I don't know how to ... | Using the idea in Aryabhata's comment, since
$$\left( 8-2x^{2}\right) ^{3}=2^{3}\left( 4-x^{2}\right) ^{3}=8
(2-x)^{3}(2+x)^{3},$$
your second equation
$$\left( 2+6x^{2}\right) \left( x+2\right) ^{3}=\left( 8-2x^{2}\right) ^{3}\tag{1}$$
is equivalent to
$$\begin{eqnarray*}
\left( 2+6x^{2}\right) \left( x+2\right) ^{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
If $a+\frac1b=b+\frac1c=c+\frac1a$ for distinct $a$, $b$, $c$, how to find the value of $abc$?
If $a, b, c$ be distinct reals such that $$a + \frac1b = b + \frac1c = c + \frac1a$$ how do I find the value of $abc$?
The answer says $1$, but I am not sure how to derive it.
| let $a+\frac{1}{b} = b+\frac{1}{c} = c+\frac{1}{a} = k$
$ab + 1 = bk, bc + 1 = ck, ca + 1 = ak \Rightarrow ab=bk-1, bc=ck-1, ca=ak-1$
$abc + c = bck = (ck-1)k = ck^2-k \Rightarrow abc + k=c(k^2-1)$
$abc + a = ack = (ak-1)k = ak^2-k \Rightarrow abc+k =a(k^2-1)$
$abc+b=abk=(bk-1)k=bk^2-k \Rightarrow abc+k=b(k^2-1)$
$c(k^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 6,
"answer_id": 1
} |
A quadratic equation $ax^2+bx+c=0$ has equal roots at $a=2c$. How could we find the sum of reciprocals of the roots of this equation?
A quadratic equation $ax^2+bx+c=0$ has equal roots at $a=2c$. How
could we find the sum of reciprocals of the roots of this equation?
I need some hints for solving this problem.
| Also, just for the fun of it, since you have a root with multiplicity two:
$$
ax^2+bx+c = (\sqrt{a}x+\sqrt{c})^2
$$
Then the root is $x^* = \pm\sqrt{\frac{c}{a}}$ and from $a = 2c$, we get $x^* = \pm\sqrt{\frac{1}{2}}$. Hence, $$
\frac{1}{x_1}+\frac{1}{x_2} = \frac{2}{x^*}=\pm 2{\sqrt{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/61690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$ Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$, where $n=0,1,2, \dots$
| Every integer is of the form $6n$ or $6n+1$ or $6n+2$ or $6n+3$ or $6n+4$ or $6n+5$ for some integer $n$. This is because when we divide an integer $m$ by $6$, we get a remainder of $0$, $1$, $2$, $3$, $4$, or $5$.
If an integer $m>2$ is of the form $6n$ or $6n+2$ or $6n+4$, then $m$ is even and greater than $2$, and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/64414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 4
} |
How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$? How can the following inequation be proven?
$$a^2 + b^2 + c^2 \ge ab + bc + ca$$
| for $a,b,c>=0$
we know $(a-b-c)^2>=0$
i.e $a^2 + b^2 + c^2 - 2 (ab + bc + ca) \ge 0$
i.e $a^2 + b^2 + c^2 \ge ab + bc + ca$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/64868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 12,
"answer_id": 7
} |
Variance in a Discrete Probability Distribution (Homework) For my Homework I have four values: $N_1, N_2, A, C$.
Two values are chosen at random without replacement.
X denotes how many N values are chosen.
$$P(x=0) = \frac16 \qquad
P(x=1) = \frac23 \qquad
P(x=2) = \frac16 $$
find the mean and variance.
I have taken t... | For any function $f$, the expectation of $f(X)$ is given by: $\newcommand{\E}{\mathrm{E}}$
$$ \E(f(X)) = \sum_{i=0}^2 \ \Pr[X = i] \cdot f(i) = \sum_{i=0}^2 \ p_i f(i) .$$
The expectation of $X$ is
$$
\E(X) = 0 \cdot p_0 + 1 \cdot p_1 + 2 \cdot p_2 = \frac{1}{6} \cdot 0 + \frac{2}{3} \cdot 1 + \frac{1}{6} \cdot 2 = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/66504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $ \int {\frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}}\mathrm dx $ I would like to evaluate: $$ \int {\frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}}\mathrm dx $$
$$ \frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}=\frac{\sqrt{1-x}+\sqrt{1+x}-2}{2(\sqrt{1-x^2}-1)} $$
The substitution $ x \rightarrow \sin(x) $ or $ \cos(x) $ can only simplify the d... | Just to give a complete answer using the method of the square. Using the easily proved equation
$$\left ( \sqrt{1+x} + \sqrt{1-x} \right )^2 = 2\left ( 1+ \sqrt{1-x^2} \right )$$
one can get
\begin{align*}
\int_{-1}^{1} \frac{\mathrm{d}x}{\sqrt{1+x} + \sqrt{1-x} + 2} &= \int_{-1}^{1} \frac{\mathrm{d}x}{\sqrt{2\left (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/67172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 5
} |
Probability question about married couples
If four married couples are arranged in a row, what is the probability
that no husband sits next to his wife?
Would it be
$1- \frac{2(4!)}{8!}$?
| Let's count the complement. If at least one couple stands together, you have $\binom{4}{1} = 4$ ways of choosing the couple, $2$ ways of arranging them, and $7$ positions in which to place them (the first person in the couple cannot stand in the last place of the line), and the remaining 6 positions can be filled in $6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/68541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
separation of variables Solve and prove Using separation of variables Solve
$$\frac{dy}{dx} = 3y +1.$$
My final answer I got was:
$$y = \pm C \frac{e^x}{3} - \frac{1}{3}.$$
But I don't how to take the derivative of that to get it back to my orig problem
| First, note that a constant divided by $3$ is just a constant, so you could rewrite what you have without the fraction. Also, plus or minus a constant is just a constant. So you could rewrite your solution as
$$y = Ce^x - \frac{1}{3}.$$
To check, you just take the derivative: $C$ is a constant, so
$$\frac{dy}{dx} = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to prove those "curious identities"? How to prove
$$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$
and
$$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$
| Define $\zeta_n = e^{2 \pi i/n}$.
Proposition
For odd integer $n \geq 1$,
\begin{align}
\prod_{k = 1}^{n-1}(\zeta_n^{k} - \zeta_n^{-k}) = n.
\end{align}
and
\begin{align}
\prod_{k = 1}^{n-1} \sin( \tfrac{2 \pi k }{n} ) = \tfrac{n}{(2 i)^{n-1}}.
\end{align}
Proof: The claimed identities follow from the identity
\beg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 4,
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Solving a problem involving continuity of a piecewise function I'm trying to solve the following problem:
Determine which $n\in\mathbb{N}$ make the following function
$f:\mathbb{R}^3\rightarrow \mathbb{R}$ continuous at the origin:
$ f(x,y,z)=\left\{
\begin{array}{cr}
\frac{(\cos^2\left(|x|+|y|\r... | The question is for $n$ integer, but we can take $a\in\mathbb R$ instead of $n$. We denote $f_a$ the corresponding function. We have for $(x,y,z)\neq (0,0,0)$
$$f_a(x,y),z)=-\frac{\sin^2(|x|+|y|)\sin (y^2+z^2)}{(x^2+y^2+z^2)^{a/2}}.$$
Now, fix a real number $m$. We have
$$f_a(x,mx,mx)=-\frac{\sin(2|mx|)\sin(2m^2x^2)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove this Pythagorean Triple? How do I prove that $(2mn, m^2 - n^2, m^2 + n^2)$ is true for $m>n>0$?
Since $m^2 + n^2$ is the hypotenuse, I applied the Pythagoren theorem:
$(2mn)^2 + (m^2 - n^2)^2 = (m^2 +n^2)^2$ and simplified it so that I would get $(m^2 + n^2)^2 = (m^2 + n^2)^2$ but I wasn't able to prove an... | Realizing that I made a silly mistake, I reevaluated the equation and:
Given the Pythagorean Triple $(2mn, m^2-n^2,m^2+n^2)$ and $m>n>0$, we can apply the Pythagorean theorem to see if they are equal:
$(2mn)^2+(m^2-n^2)^2 = (m^2+n^2)^2$
$4m^2n^2 + (m^4-2m^2n^2+n^4) = (m^2+n^2)^2$
$m^4+2m^2n^2+n^4=(m^2+n^2)^2$
$(m^2+n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/70808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Cauchy-Schwarz inequality and three-letter identities (exercise 1.4 from "The Cauchy-Schwarz Master Class") Exercise 1.4 from a great book The Cauchy-Schwarz Master Class asks to prove the following:
For all positive $x$, $y$ and $z$, one has
$$x+y+z \leq 2 \left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + \frac{z^2}{x+y}\righ... | $(x+y+z)^2 = \left(x\sqrt{\frac{y+z}{y+z}} +y\sqrt{\frac{x+z}{x+z}}+z\sqrt{\frac{x+y}{x+y}}\right)^2\leq\left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + \frac{z^2}{x+y}\right)(y+z+x+z+x+y)$. You will get it. I didn't read that book, but I believe Srivatsan is right.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/71509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 0
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Divisibility question Let $r$ be an integer greater than $2$.
Is there a simple way of showing that $2^r$ divides
$\left(\begin{array}{c}
{2}^{r-2} \\
k
\end{array}\right) 2^{2k}$
but it does not divide
$\left(\begin{array}{c}
2^{r-l} \\
k
\end{array}\right) 2^{2k}$ for $l>2$ and for all $k$ greater than
$0$ ... | I don't know off-hand about the binomial coefficients, but here's a proof of the order of $5$.
For $r=3$, the order of $5$ is $2$, which agrees with what we want. So assume that $r\gt 3$.
Assume that the order of $5$ has order $2^{r-2}$ modulo $2^{r}$. The order of $5$ modulo $2^{r+1}$ is a multiple of $2^{r-2}$. It ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Interscholastic Mathematics League Senior B #12 Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$.
So here is what I have done so far. I got two of the roots to be zero and 4 since $x^3(x+4)+(6x^2+1004x+1001)=0$. But for the other two roots by using the quadratic formula I get $$\fra... | $4$ is not a root by the rational root theorem and $0$ is not a root since the polynomial is $1001$ at $x=0$. We have $(x+1)^4+1000(x+1)=0$, so one real root is $-1$. Taking that out we have a sum of cubes, $(x+1)^3+10^3=0$, which factors as $(x+11)((x+1)^2-10(x+1)+100)=0$. Eliminating the real root $-11$ we are lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Formula for completing the square? My math teacher said that this was the formula for completing the square.
Original function: $$ax^2 + bx + c$$
Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$
However, using this formula I'm not getting the same answers that I would get just by determining ... | I usually find that completing the square by hand for each example is better than just using the formula.
For your example,
$$4x^2 + 4x + 5 = 4\left(x^2 + x + \frac{5}{4}\right).$$
This is of the form $x^2 + x + \mathrm{const}$, so you to find a number $a$ such that $(x+a)^2 = x^2 + x + \mathrm{const}$. The solution i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Irreducible polynomial which is reducible modulo every prime
How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$?
For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not solutions of $x^4+1=0$ then if it is reducible the factors are ... | When $p=2$ then just note $x^4+1=(x+1)^4$.
Now if $p$ is odd then $8\mid p^2-1 \implies x^4+1\mid x^{p^2-1}-1\mid x^{p^2}-x$. Let $a$ be a root of $x^4+1$ in some extension of $\mathbb F_p$. So, $[\mathbb F_p(a):\mathbb F_p]=4$ if we let $x^4+1$ is irreducible over $\mathbb F_p$. But from $x^4+1\mid x^{p^2}-x$ we can s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 3,
"answer_id": 2
} |
Summation Identity: $\sum_{i=1}^ni^3 = \left( \frac{n(n+1)}{2} \right)^2$ I have to prove:
$$\sum\limits_{i = 1}^n i^3 = \Bigg( \frac{n(n+1)}{2}\Bigg)^2$$
Using the following:
$$n^3 = 6 {n \choose 3} + 6 {n \choose 2} + n \quad \forall n \in \mathbb{N}$$
My work is that first I substitute $n^3$ for $6 {n \choose 3} + 6... | It’s just algebra $-$ tedious, but straightforward:
$$\begin{align*}
6\binom{n+1}4&+6\binom{n+1}3+\binom{n+1}2\\
\strut\\
&=\frac{6(n+1)n(n-1)(n-2)}{4!}+\frac{6(n+1)n(n-1)}{3!}+\frac{(n+1)n}2\\
&=n(n+1)\left(\frac{(n-1)(n-2)}{4}+(n-1)+\frac12\right)\\
&=\frac{n(n+1)}4\big((n-1)(n-2)+4(n-1)+2\big)\\
&=\frac{n(n+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
$1+1/(2+3)+1/(4+5+6)...$ What is the value of the sum of the series
$$\frac{1}{1}+\frac{1}{2+3}+\frac{1}{4+5+6}+\dotso\;?$$
And this:
$$\frac{1}{1}+\frac{1}{2\cdot3}+\frac{1}{4\cdot5\cdot6}+\dotso\;?$$
| Note $a+(a+1)+\cdots+(b-1)+b= (b-a+1)(b+a)/2$. Set $b=n(n+1)/2,a=n(n-1)/2+1$:
$$(\circ)=\sum_{n=1}^\infty \left((n(n+1)/2-[n(n-1)/2+1]+1)\cdot\frac{n(n+1)/2+[n(n-1)/2+1]}{2}\right)^{-1}$$
$$=\sum_{n=1}^\infty\frac{2}{n(n^2+1)}=\sum_{n=1}^\infty\left(\frac{2}{n}-\frac{1}{n+i}-\frac{1}{n-i}\right)=\sum_{n=1}^\infty\int_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
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Converting base 10 fractions into other bases How do you convert
$14\frac{8}{13}$
into base 3?
I was able to convert $\frac{3}{7}$ into base 3 by constantly multiplying by 3 and dividing the numerator by denominator until I finally got a repetition, but this method doesn't seem to work for $14\frac{8}{13}$.
The corr... | Just like you convert a fraction to decimal. $8\cdot 3=24=13(1)+11,$ so the first digit (ternit?) is $1. \ \ 11\cdot 3=33=13(2)+7,$ so the second digit is $2. \ \ 3\cdot 7=21=13(1)+8$ and so on. I get $112.\overline{121}_{3}$
Alternately you can notice that $\frac{8}{13}=\frac{16}{3^3-1}$, so the repeat is $3$ digi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Need help finding smallest value of $x^2 + y^2$ I need to find the smallest value of $x^2 + y^2$ with the restriction $2x + 3y = 6$. This chapter focuses on the vertex formula.
| For fun, we give a couple of solutions that are not the intended ones. The solutions are very similar, but the first is expressed algebraically, while the
second brings in the geometry.
$1$) Note that
$$(2x+3y)^2+(3x-2y)^2=13(x^2+y^2).$$
Thus, given that $2x+3y=6$,
$$13(x^2+y^2)= 36+(3x-2y)^2.$$
If we can manage to m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
What is the integral of $x(1-x)^8$? I want to find the integral of $x (1-x)^8$. How do I go about this? For example, which rule do I use from http://integral-table.com ? Thanks!
| $$x(1-x)^8 = x-8 x^2+28 x^3-56 x^4+70 x^5-56 x^6+28 x^7-8 x^8+x^9$$
so
$$\begin{array}{cl}
&& \int x(1-x)^8 \,dx \\
&=& \int x-8 x^2+28 x^3-56 x^4+70 x^5-56 x^6+28 x^7-8 x^8+x^9 \,dx \\\\
&=& \int x \,dx - 8 \int x^2 \,dx +28 \int x^3 \,dx - 56 \int x^4 \,dx + 70 \int x^5 \,dx - 56 \int x^6 \,dx + 28 \int x^7 \,dx -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Square Roots of Complex Number $3-4i$
What I did
$z^2=3-4i$
$(a+bi)^2 = 3-4i$
$a^2-b^2+2abi = 3-4i$
Then got 2 simultaneous equations
$a^2-b^2=3$ and $2ab=-4$
Solve for $a^2$ in 1st equation: $a^2=3+b^2$
Subbed into 2nd equation to power of 2
$(3+b^2)b^2=4$
$b^4+3b^2-4=0$
Is there a better way than below? Solving powe... | The high school way:
If $(x+iy)^2=3-4i$, expanding the l.h.s. and identifying the real and imaginary parts, you obtain the system of equations
$\;\begin{cases}
x^2-y^2=3,\\2xy=-4.
\end{cases}$
Now, you can also write that $|x+iy|^2=|3-4i|$, i.e. $\;x^2+y^2=5$. Therefore you obtain the linear system in $x^2$ and $y^2$:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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alternative proof to induction, binomial sum identity
$\displaystyle \sum_{q=0}^{k} \begin{pmatrix}n-1+q\\ n-1 \end{pmatrix} = \begin{pmatrix}n+k\\n \end{pmatrix} $
induction
$\begin{pmatrix}n \\ k \end{pmatrix} := \frac{n!}{(n-k)!k!}$
Beginning of induction : $k=0 \rightarrow \begin{pmatrix} n-1 \\ n-1 \end{pma... | $\binom{n+k}{n}$ is the number of ways of choosing $n$ objects from $n+k$ possibilities.
In order to select $n$ objects out of $n+k$ possibilities, we can also proceed as follows: number the objects from $1$ throught $n+k$. First, pick the object with the largest number that you will select; it must be numbered somewh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Solve $8^{x+1} = 32 \cdot\sqrt2$ without $\log$ I need help solving the equation $8^{x+1} = 32 \cdot\sqrt2$. The obvious answer is to use log, but that is reserved for the next section. The example given for this section of questions is:
$4^x = 8$
$(2^2)^x = 2^3$
$2x = 3$
$x = \dfrac{3}{2}$
The example looks obvious an... | The idea is that everything in sight is a power of $2$.
$$
\begin{eqnarray*}
8^{x+1} &=& 32 \sqrt{2} \\
(2^3)^{x+1} &=& 2^5\cdot 2^{1/2} \\
2^{3(x+1)} &=& 2^{5 + 1/2}\\
\end{eqnarray*}
$$
Since both sides are expressed as a power of the same base, we can set the exponents equal:
$$
\begin{eqnarray*}
3(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
how to solve $\pm y \equiv 2x+1 \pmod {13}$ with Chinese remainder theorem or iterative method? how to solve $\pm y \equiv 2x+1 \pmod {13}$ with Chinese remainder theorem or iterative method?
It comes from solving $x^2+x+1 \equiv 0 \pmod {13}$ (* ) and background is following:
13 is prime. (* ) holds under Euclidean l... | Either, $\pm y-1$ is divisible by 2, so you divide by 2, or $\pm y+12$ is divisible by 2, so divide this by 2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/85814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is this proof about Mersenne numbers acceptable? I want to prove following property of Mersenne numbers :
If $p > 3$ then $M_p\equiv 1 \pmod {6\cdot p}$
So, according to Fermat's Little Theorem we may write :
$2^p\equiv 2 \pmod p \Rightarrow 2^p-2=a\cdot p \Rightarrow 2^p-1=a\cdot p +1 \Rightarrow$
$\Rightarrow 2^p-1... | Note that $M_p= 2^p -1 = (2-1)(2^{p-1} + 2^{p-2} + \dots + 2^2 + 2^1 + 1)$. If $p>3$ then modding out by $2^3=8$ leaves $2^2+2^1+1 =7$ behind. Also $M_p = 2^p-1$ and $p$ is odd, so $2^p \equiv 2 \mod 3$, which means $M_p \equiv 1\mod 3$. The Chinese Remainder Theorem then applies and shows that $M_p \equiv 7 \mod (3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluate the limit of $\lim_{x\to\infty}\frac{x - \sqrt{x^2+5x+2}}{x-\sqrt{x^2+0.5x + 1}}.$ $$\lim_{x\to\infty}\frac{x - \sqrt{x^2+5x+2}}{x-\sqrt{x^2+0.5x + 1}}.$$
Can't find a means to resolve. The answer is $10$ by graphing.
| We give two solutions. The first uses a standard trick that you have probably already seen. The second uses the definition of the derivative.
You are familiar with the identity
$$(\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})=u-v, \qquad \qquad (\ast)$$
at least in its incarnation $(a-b)(a+b)=a^2-b^2$. We will use $(\ast)$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"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.