Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Max and Min$ f(x,y)=x^3-12xy+8y^3$ First I solved $f_y=0$ then plugged in my variable into $f_x$ to get an output and then plugged that output back into $f_y$ to get a point and did this again for $f_x$ I think I screwed up on the $(-7,\sqrt{3})$
$f_y=-12x+24y^2$
$-12x+24y^2=0$
$-12x=24y^2$
$x=2y^2$
$f_x(2y^2,y)=3(2y^2... | $$f_x=3x^2-12y$$
$$f_y=24y^2-12x$$
Now we let $\nabla f(x,y)=(f_x,f_y)=0$ which gives us the following:
$$
\left\{
\begin{array}{c}
f_x=3x^2-12y=0 \\
f_y=24y^2-12x=0
\end{array}
\right.
$$
Solving it we have
$$
\left\{
\begin{array}{c}
x=2y^2 \\
12y^4-12y=12y(y^3-1)=0
\end{array}
\right.
$$
Where $y=0\lor y=1$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2158562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the number of roots of the equation $3^{|x|}-|2-|x||=1$
Find the number of roots of the equation $3^{|x|}-|2-|x||=1$
My working:
Let $t$ be any positive real number.
$3^{t}-|2-t|=1$
Case 1:
$t<2$
$3^{t}-2+t=1$
$3^{t}+t=3$
Case 2:
$t>2$
$3^{t}+2-t=1$
$3^t+1=t$
Now I don't know how to proceed... | $f(x)=3^{|x|}-|2-|x||$ is an even function ,so
$f(x)=1$ has symmetric roots (like $x=\pm a$)
so ,solve the equation for $x \geq 0 $ and then add negative of root(s)
$$3^{|x|}-|2-|x||=1, x \geq 0 \\ 3^x-|2-x|=1 \to
\begin{cases}3^x-(2-x)=1 & x <2\\3^x+(2-x)=1 & x >2\end{cases}\\
\begin{cases}3^x=3-x & x <2\\3^x=-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2158816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $f_k(x)=\frac{1}{k}\left (\sin^kx +\cos^kx\right)$, then $f_4(x)-f_6(x)=\;?$ I arrived to this question while solving a question paper. The question is as follows:
If $f_k(x)=\frac{1}{k}\left(\sin^kx + \cos^kx\right)$, where $x$ belongs to $\mathbb{R}$ and $k>1$, then $f_4(x)-f_6(x)=?$
I started as
$$\begin{align... | ‘Tis I have a better solution to solve this math problem...
Let $a = \cos^2 x$ and $b = \sin^2 x,$ so $a + b = 1.$ Then
[(a + b)^2 = a^2 + 2ab + b^2 = 1,]so $a^2 + b^2 = 1 - 2ab.$ Also,
[(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3 = 1,]so
\begin{align*}
a^3 + b^3 &= 1 - (3a^2 b + 3ab^2) \\
&= 1 - 3ab(a + b) \\
&= 1 - 3ab.
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2159136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How do I find complex function knowing its real part and value in zero? For example I have a complex function $f(z)$, $z = x + iy$.
I know that $\operatorname{Re} f(z) = x^3 + 6 x^2 y + A x y^2 - 2 y^3$
and that $f(0) = 0$.
What should I do to find $f(z)$?
| First method: Cauchy-Riemann equations. The function $f$ is assumed analytic. Therefore, there is an open set around zero where it is holomorphic. The Cauchy-Riemann equations write
\begin{equation}
\frac{\partial\, \text{Im} f}{\partial y}(x+\text{i}y) = \frac{\partial\, \text{Re} f}{\partial x}(x+\text{i}y) = 3x^2 + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2159317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Landing on Consecutive Spaces with Dice Rolls Imagine you are continuously rolling one six-sided fair die, and moving a token the amount on the die on a board with 1000 spaces. Assuming the token starts at zero, is there a general formula that describes the probability you will land on any three consecutive spaces? For... | Let's start by defining $r(n)$ to be the probability that the sum is ever exactly $n$.
Let's express the probability that the sum will ever be $a$, $a+1$, or $a+2$ using $r$. Let's call that probability $P$.
For a set of positive integers $S$, define $p_s$ to be the probability that the sum will ever be in $S$.
Usin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2161678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
diophantine equation $x^2+y^2=3$ what is the solution of this diophantine equation $x^2+y^2=3$ ? with x and y are the rationnals number I have one solution : $x=\frac{a}{c}$ and $y=\frac{b}{c}$ a, b and c are the positive integer
I found a solution by doing a reasoning modulo 3
| Transform your search for rational solutions into a search for positive integer solutions by setting $x=a/c$, $y=b/c$, so the equation becomes
$$
a^2+b^2=3c^2
$$
Looking at this modulo $3$, we conclude that $a\equiv0\pmod{3}$ and $b\equiv{0}\pmod{3}$ (prove it). Therefore $a=3a_1$, $b=3b_1$ and so
$$
3a_1^2+3b_1^2=c^2
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2163153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Simplify this trigonometric equation (identities) Please simplify these trigonometric identities and describe step by step (wording).
$\frac{\sec \theta-\csc \theta}{1-\cot\theta}$
| Let $A = \frac{\sec\theta - \csc\theta}{1-\cot\theta}$
$A = \frac{\frac{1}{\cos\theta}-\frac{1}{\sin\theta}}{1-\frac{\cos\theta}{\sin\theta}}$
Now we make common denominators like so:
$A = \frac{\frac{\sin\theta}{\sin\theta \cdot \cos\theta}-\frac{\cos\theta}{sin\theta\cdot \cos\theta}}{\frac{\sin\theta}{\sin\theta}-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2166247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Infinitely many $n$ such that $n, n+1, n+2$ are each the sum of two perfect squares. Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are all the sum of two perfect squares. Induction does not seem to be yielding any results.
| n = 0 is a trivial solution as 0 = $0^2 + 0^2, 1 = 1^2 + 0^2, 2 = 1^2 + 1^2$. Consider the pell equation $x^2 - 2y^2 = 1$, which has infinitely many solutions. Take $n = x^2 -1$. This implies $ n = y^2 + y^2$, $n+1 = x^2 + 0^2$ and $n+2 = x^2 + 1^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2170494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
Does this system of equations has a solution? Let $R$ be an integral domain with identity. Let $A$ be an $n\times n$ matrix with entries from $R$. Assume that $A$ has the following properties:
(1) Every row of $A$ has exactly one $1$ and one $-1$.
(2) Every column of $A$ has exactly one $1$ and one $-1$.
(3) All other ... | No: Consider the following system of equations (we work in $R = \mathbb{Z}$):
$$\begin{pmatrix}
1 & -1 & 0 & 0\\
-1 & 1 & 0 & 0\\
0 & 0 & 1& -1\\
0 & 0 & -1 & 1
\end{pmatrix}\cdot
\begin{pmatrix}
x_1 \\
x_2\\
x_3\\
x_4
\end{pmatrix}
=
\begin{pmatrix}
1\\
2\\
-1\\
-2
\end{pmatrix}$$
Then we would have that $1 = x_1 - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine whether $\int^{\infty}_{2} \frac{1}{x\sqrt{x^2-4}} dx$ converges. Determine whether $$\int^{\infty}_{2} \frac{1}{x\sqrt{x^2-4}} dx$$ converges.
Doing some rough work, I realize that this function near $\infty$, behaves like $$\frac{1}{x\sqrt{x^2}} = \frac{1}{x^2}$$
I know this function converges, but I am hav... | Let $x-2=u$. Then $dx=du$. The integral becomes \begin{align*} \int_0^\infty \frac{du}{(u+2)\sqrt{(u+2)^2-4}}&=\int_0^\infty \frac{du}{(u+2)\sqrt{u^2+4u}}=\int_0^\infty \frac{du}{\sqrt{u^3+6u^2+8u}} \\ \\ &= \underbrace{\int_0^1 \frac{du}{\sqrt{u^3+6u^2+8u}}}_{=I_1} +\underbrace{\int_1^\infty \frac{du}{\sqrt{u^3+6u^2+8... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2171854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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Proving: $\left(a+\frac1a\right)^2+\left(b+\frac1b\right)^2\ge\frac{25}2$
For $a>0,b>0$, $a+b=1$ prove:$$\bigg(a+\dfrac{1}{a}\bigg)^2+\bigg(b+\dfrac{1}{b}\bigg)^2\ge\dfrac{25}{2}$$
My try don't do much, tough
$a+b=1\implies\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{ab}$
Expanding: $\bigg(a+\dfrac{1}{a}\bigg)^2+\bigg(b+\dfra... | Calculus will also work. Let $f(a)=(a+\frac {1}{a})^2 +(b+\frac {1}{b})^2$ with $b=1-a$ and, WLOG, $a\geq b.$ So $1/2\leq a<1.$
Since $db/da=-1$ we have $$f'(a)=2\left(a+\frac {1}{a}\right) \left(1-\frac {1}{a^2}\right)+2\left(b+\frac {1}{b}\right) \left(-1+\frac {1}{b^2}\right)=$$ $$=2(a-b)+2\left(\frac {1}{b^3}-\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2172634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Prove by induction that $n^4+2n^3+n^2$ is divisible by 4
I'm trying to prove by induction that $n^4+2n^3+n^2$ is divisible by 4.
I know that P(1) it's true. Then $ n=k, P(k):k^4+2k^3+k^2=4w$ it's true by the hypothesis of induction.
When I tried to prove $n=k+1, P(k+1):(k+1)^4+(k+1)^3+(k+1)^2 = 4t$,
$$k^4+4k^3+6k^2+... | As an alternative to proving via induction, consider the following hints:
Hint #1: $n^4 + 2n^3 + n^2 = n^2(n^2 + 2n + 1) = \bigg(n(n+1)\bigg)^2$
Hint #2: The parity of $n$ is different from that of $n+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2173455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Evaluating the following summation $\require{cancel}$
Evaluate $$\sum_{n=2}^{\infty}\ln\left(\frac{n^2-1}{n^2}\right)$$
My procedure:
$$\sum_{n=2}^{\infty}\ln\left(\frac{n^2-1}{n^2}\right)=\sum_{n=2}^{\infty}(\ln\left(n-1\right)-\ln(n)+\ln(n+1)-\ln(n))$$
If we evaluate a few terms:
$$\sum_{n=2}^{m}(\ln\left(n-1\right)-... | Another way would be to consider the exponentiation. Observe
\begin{align}
\exp\left(\sum^N_{n=2}\ln\frac{n^2-1}{n^2} \right)=&\ \prod^N_{n=2}\frac{n^2-1}{n^2} = \prod^N_{n=2}\frac{(n-1)(n+1)}{n \cdot n}\\
=&\ \frac{(2-1)(2+1)}{2\cdot 2}\cdot \frac{(3-1)(3+1)}{3\cdot 3}\cdots\frac{(N-2)N}{(N-1)\cdot (N-1)}\frac{(N-1)(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2175317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Limit of an indeterminate form with a quadratic expression under square root The problem is:
$$ \lim_{x\to0}\frac{\sqrt{1+x+x^2}-1}{x} $$
So far, I've tried substituting $\sqrt{1+x+x^2}-1$ with some variable $t$, but when $x\to0$, $t\to\sqrt{1}$.
I have also tried to rationalize the numerator, and applied l'hospital... | $$\lim _{ x\to 0 } \frac { \sqrt { 1+x+x^{ 2 } } -1 }{ x } \cdot \frac { \sqrt { 1+x+x^{ 2 } } +1 }{ \sqrt { 1+x+x^{ 2 } } +1 } =\lim _{ x\to 0 } \frac { x+{ x }^{ 2 } }{ x\sqrt { 1+x+x^{ 2 } } +1 } =\lim _{ x\to 0 } \frac { 1+x }{ \sqrt { 1+x+x^{ 2 } } +1 } =\frac { 1 }{ 2 } $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2176506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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How to find the numbers of sets of positive numbers $\{a,b,c\}$ such that $(a)(b)(c) =2^{4}3^{5}5^{2}7^{3}$? How to find the numbers of sets of positive numbers $\{a,b,c\}$ such that $(a)(b)(c) =2^{4}3^{5}5^{2}7^{3}$ ?
$$\binom{4+2}{2} \binom{5+2}{2} \binom{2+2}{2} \binom{3+2}{2}$$
This would give the number of tuple... | Let $S = \{(a,b,c) \colon abc = 2^4 3^5 5^2 7^3\}$. Let's split it into few smaller sets. $S_1 = \{(a,b,c) \in S \colon a \neq b;\, b \neq c;\, a \neq c\}$, $S_2 = \{(a,b,c) \in S \colon a=b;\, b \neq c\}$ and let $S_3, S_4$ be similar to $S_2$ only with equal pairs $a=c$ and $b=c$. It's easy to notice that $S = S_1 \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2177524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Determine the eigenvector and eigenspace and the basis of the eigenspace The yellow marked area is correct, so don't check for accuracy :)
$A=\begin{pmatrix} 0 & -1 & 0\\ 4 & 4 & 0\\ 2 & 1 & 2
\end{pmatrix}$ is the matrix.
Characteristic polynomial is $-\lambda^{3}+6\lambda^{2}-12\lambda+8=0$
The (tripple) eigenva... | From your calculation you have that all eigenvectors are of the form $\begin{pmatrix}
x\\
-2x\\
z
\end{pmatrix} = x\begin{pmatrix}
1\\
-2\\
0
\end{pmatrix} + z\begin{pmatrix}
0\\
0\\
1
\end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2177817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
$\sum_{n=1}^{N}\sin(x)\cos((2n-1)x) = \frac{1}{2}\sin(2Nx)$ I want to prove the following by induction $$\sum_{n=1}^{N}\sin(x)\cos((2n-1)x) = \frac{1}{2}\sin(2Nx)$$
here's my trial:
Base Step
N = 1 we have that $LHS = \sin(x)\cos(x)$ and $RHS = \frac{1}{2}\sin(2x) = \sin(x)\cos(x)$. Hence it works for N=1.
Inductive St... | So in total we have $$\sum_{n=1}^{N+1}\sin(x)\cos((2n-1)x) =\frac{1}{2}\sin(2Nx)+\frac{1}{2}\cos(2Nx)\sin(2x)-\sin(2Nx)\sin^2(x)$$ and rearranging we have $$\sum_{n=1}^{N+1}\sin(x)\cos((2n-1)x) =\sin(2Nx)\left(\frac{1}{2}-\sin^2(x)\right)+\frac{1}{2}\cos(2Nx)\sin(2x) $$ and again $$\sum_{n=1}^{N+1}\sin(x)\cos((2n-1)x) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2179633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does this radical ${a-1\over 2}\left({\sqrt[3]a+1\over \sqrt[3]a-1}-1\right)=...$ hold? Consider the nested radical $(1)$
$${a-1\over 2}\left({\sqrt[3]a+1\over \sqrt[3]a-1}-1\right)=2\sqrt[3]a+\left[(a^2-7a+1)+(6a-3)\sqrt[3]a+(6-3a)\sqrt[3]{a^2}\right]^{1/3}\tag1$$
An attempt:
$x=\sqrt[3]a$
$$\left({a-1\over 2}\left({x... | $y^3-1=(y-1)(y^2+y+1)$ ... let $y=\sqrt[3] a$. So I guess your LHS is
\begin{eqnarray*}
\frac{a-1}{2} \left( \frac{\sqrt[3] a+1 }{\sqrt[3] a-1}-1 \right) = (\sqrt[3] a)^2+\sqrt[3] a+1
\end{eqnarray*}
Now subtract $2 \sqrt[3] a$ from both sides and then cube both sides ... you will get the second term of the RHS ... bu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2180701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Noob question about $\int \frac{1}{\sin(x)}dx$ I manually integrate $\int \frac{1}{\sin(x)}dx$ as $$\int \frac{\sin(x)}{\sin^{2}(x)}dx = -\int \frac{1}{\sin^{2}(x)}d\cos(x) = \int \frac{1}{\cos^{2}(x) - 1}d\cos(x).$$
After replacing $u = \cos(x)$,
$$\int \frac{1}{u^{2} - 1}du = \int \frac{1}{u^{2} - 1}du = \frac {1} ... | If you are considering antiderivatives as functions on connected domains $D$ that are open in $\mathbb{C}$ -- suitably chosen so that we don't have to worry about multivaluedness of the logarithm -- then you should still be okay. It's still true that the functions $z \mapsto \frac1{2} \ln \left(\frac{1 - \cos(z)}{1 + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2181582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Finding a basis for a set of $2\times2$ matrices
Find a basis for $M_{2\times2}$ containing the matrices $$\begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix}$$ and $$\begin{pmatrix} 1 & 1 \\ 3 & 2 \end{pmatrix}$$
I know that every $2\times2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ can be written as:
$$a\begi... | Here's a systematic approach: to begin, find a basis for $\Bbb R^4$ containing the vectors $(1,1,2,3)$ and $(1,1,3,2)$. Following the method that I outline here, we find that $\{(1,1,2,3),(1,1,3,2),(1,0,0,0),(0,1,0,0)\}$ is a basis for $\Bbb R^4$. Correspondingly,
$$
\left\{\pmatrix{1&1\\2&3},
\pmatrix{1&1\\3&2},
\pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2184690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Proof involving integer squares and parity Original proposition:
$a$, $b$ and $c$ are integers. Show that if $a^2+b^2=c^2$, at least
one of $a$ and $b$ is an even number.
My attempt:
Attemplting proof by contradiction:
Show that there exist a contradiction between the claim that $a^2+b^2=c^2$
...and $a=2k+1, b=2k+1... | Assume by way of contradiction $a$ and $b$ are odd, as you say we get that $$4(k^2+k+l^2+l)+2=c^2.$$ We have that $c^2$ is even thus $c$ is even, so $c=2m$ for some $m$. We get $$4(k^2+k+l^2+l)+2=4m^2.$$
divide both sides by two:
$$2(k^2+k+l^2+l)+1=2m^2$$
This is a contradiction since LHS is odd and RHS is even.
| {
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"url": "https://math.stackexchange.com/questions/2185759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Using L'Hopital Rule, evaluate $\lim_{x \to 0} {\left( \frac {1} {x^2}-\frac {\cot x} {x} \right)}$ Using L'Hopital Rule, evaluate $$ \lim_{x \to 0} {\left( \frac {1} {x^2}-\frac {\cot x} {x} \right)}$$
I find this question weired.
If we just combine the two terms into one single fraction, we get$$\lim_{x \to 0} {\fra... | first :
$$\lim_{x\to 0} \frac{\sin x-x}{x^3}=\frac{-1}{6} \tag{1}$$
$$\lim_{x\to 0} \frac{1-\cos x}{x^2}=\frac{3}{6} \tag{2}$$
$$\lim_{x\to 0} \frac{x}{\sin x}=1 \tag{3} $$
$$\frac{1}{x^2}-\frac{\cot x}{x}=\frac{\sin x-x}{x^3}\times \frac{x}{\sin x}+\frac{1-\cos x}{x^2}\times \frac{x}{\sin x}$$
$$L=\lim_{x\to 0} \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2186061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 5
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Find a prime number p that is simultaneously expressible in the forms $x^2 + y^2, u^2 + 2v^2$, and $r^2 + 3s^2$. background: From Burton, Number Theory 9.3#8 with a hint: (−1/p) = (−2/p) = (−3/p) = 1.]
Based on the previous problem 6:
$n^2+1, n^2+2$, or $n^2+3$ have solutions when $n^2\equiv-i\mod p$ $(\frac{-i}{p})=1... | Let $p$ be an odd prime. Using Legendre symbol rules, one can deduce that
\begin{equation*}
\begin{aligned}
p &\equiv 1 \pmod 4, \\
p &\equiv 1,7 \pmod 8, \\
p &\equiv 1,11 \pmod{12}. \\
\end{aligned}
\end{equation*}
The above system has no solution unless $p$ is equivalent to $1$ modulo $4,8,12$. In this case, $$p \eq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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prove that $\lim_{x \to 1}\frac{x^{2}-x+1}{x+1}=\frac{1}{2}$ Please check my proof
Let $\epsilon >0$ and $\delta >0$
$$0<x<\delta \rightarrow \frac{x^{2}-x+1}{x+1}-\frac{1}{2}<\epsilon $$
$$\frac{2x^{2}-2x+2-x+1}{2x+2}<\epsilon $$
$$\frac{2x^{2}-3x+1}{2x+2}<\epsilon $$
since $\frac{2x^2-3x+1}{2x+2} <\frac{2x^{2}-3x+... | This function is defined at continuous at x = 1.
(It would be a different problem is we were looking for the limit as x approaches -1)
Unless your professor is demanding epsilon-delta proofs for the practice, you know that the limit of a defined continuous f as x approaches 1 is
$f(1) = \frac{1^2 - 1 + 1}{1 + 1} = \f... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing that a function from $\mathbb{R}^2 \to\mathbb{R}$ is bounded
Show that
$$ f(x,y)= \begin{cases} \dfrac{xy^2}{x^2+y^4} & (x,y) ≠ (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$
is bounded.
I thought about splitting it up into different cases like $x<y$ but it turned out to be too many and I could not cover all ... | The insight here is that the fraction is of the form $\frac{ab}{a^2+b^2}$, with $a=x$ and $b=y^2$.
\begin{align*}
(x-y^2)^2 &\geq 0 \\
\implies x^2 - 2xy^2 + y^4 &\geq 0 \\
\implies x^2 + y^4 &\geq 2xy^2 \\
\implies\frac{xy^2}{x^2 + y^4} &\leq \frac{1}{2}
\end{align*}
You can also apply the arithmetic mean-geometri... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Plotting complex arcs on an Argand diagram What would $arg(\frac{z-4i}{z+2i}) = \frac{\pi}{4}$ look like on an Argand diagram and why?
I don't know how to check my answer because Desmos does not support complex numbers.
| We have a Möbius transformation
$$z'=\frac{z-4i}{z+2i}$$
which maps circles to circles or straight lines, and straight lines to circles or straight lines. We are looking for the shape of the set of complex numbers which will be mapped to a straight line of slope $1$. We suspect that the shape is that of a circle.
Let ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2194191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the co-efficient of $x^{18}$ in the expansion of $(x+1)(x+2)...(x+10)(2x+1)(2x+3)...(2x+19)$.
Find the co-efficient of $x^{18}$ in the expansion of
$$(x+1)(x+2)...(x+10)(2x+1)(2x+3)...(2x+19)$$
What I've done :
$$
(x+1)(x+2)...(x+10)(2x+1)(2x+3)...(2x+19)
\\
=\frac{(2x+1)(2x+2)(2x+3)...(2x+20)}{2^{10}}
$$
I ca... | Now to get a term with $x^{18}$ you have to choose $18$ of the $2x$ terms and two of the constant terms, so your term in the numerator is $$(2x)^{18}\sum_{i=1}^{19}\sum_{j=i+1}^{20} ij$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simplification of $\binom{50}{0}\binom{50}{1} + \binom{50}{1}\binom{50}{2}+⋯+\binom{50}{49}\binom{50}{50}$ There was a post on this web site an hour ago asking for the sum of
\begin{equation*}
\binom{50}{0}\binom{50}{1} + \binom{50}{1}\binom{50}{2}+⋯+\binom{50}{49}\binom{50}{50}
\end{equation*}
expressed as a single bi... | There is another proof for this problem that I am mentioning here :
Since :
$$
\binom{50}{0}\binom{50}{1} + \binom{50}{1}\binom{50}{2}+⋯+\binom{50}{49}\binom{50}{50} \\
\qquad = \binom{50}{0}\binom{50}{49} + \binom{50}{1}\binom{50}{48}+⋯+\binom{50}{49}\binom{50}{0} \\$$
Consider the product :
$$(1+x)^{50} \times (1+x)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2195269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Vectors $v_1,v_2,v_3$ belong to vector space $\mathbb{Z}^{3}_{7}$ What's dimension of the subspace $U_7$?
$v_1=\begin{pmatrix} 1\\ 3\\ 5 \end{pmatrix} , v_2=\begin{pmatrix}
4\\ 5\\ 6 \end{pmatrix}, v_3 = \begin{pmatrix} 6\\ 4\\ 2
\end{pmatrix}$ are vectors of the vector space $\mathbb{Z}^{3}_{7}$,
over the fie... | We could have simply reduced the matrix by rows (with rows = the given vectors):
$$\begin{pmatrix}
1&3&5\\
4&5&6\\
6&4&2\end{pmatrix}\stackrel{R_2-4R_1,\,R_3-6R_1}\longrightarrow\begin{pmatrix}
1&3&5\\
0&0&0\\
0&0&0\end{pmatrix}\implies$$
The second and third vectors are linearly dependent on the first one , and in fac... | {
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"timestamp": "2023-03-29T00:00:00",
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If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$ Let $a$, $b$ and $c$ be positive numbers such that $a+b+c=abc$. Prove that:
$$\frac{1}{7a+b}+\frac{1}{7b+c}+\frac{1}{7c+a}\leq\frac{\sqrt3}{8}$$
I tried C-S:
$$\left(\sum_{cyc}\frac{1}{7a+b}\right)^2\leq\sum_{cyc}\frac{1}{(ka+mb+c)(7a+b)^2}\sum_{... | Not a proof just a simplification for the proof wich already exists :
We prove the following inequality :
Let $a,b,c>0$ such that $abc=a+b+c$ and $a\geq b \geq c$ then we have :
$$\sum_{cyc}\frac{1}{7a+b}\leq \sum_{cyc}\frac{1}{7b+a}$$
Proof :
We work with the following equivalent :
Let $a,b,c>0$ and $a\geq b \ge... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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proof by induction (summations) I'm trying to use induction to prove that
$$\frac{1}{(1)(2)}+\frac{1}{(2)(3)}+...+\frac{1}{(n)(n+1)}=1-\frac{1}{n+1}.$$
This is what I have so far:
Base Case: $n=1$. We get $$\sum_{i=1}^{k} \frac{1}{(i)(i+1)} = 1-\frac{1}{(k+1)} \implies \frac{1}{2} = \frac{1}{2}.$$
End Goal (what we wa... | You are almost there. The trick goes as by writing
$$\frac{1}{(k+1)(k+2)}=\frac{1}{k+1}-\frac{1}{k+2}$$ so that
$$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}=1-\frac{1}{k+1}+\frac{1}{k+1}-\frac{1}{k+2}=1-\frac{1}{k+2}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Explain why twice the sum $\binom{12}{0} + \binom{12}{1} + \binom{12}{2} + \binom{12}{3} + \binom{12}{4} + \binom{12}{5}$ is $2^{12}-\binom{12}6$ Can someone explain how is the RHS concluded? I did with sample numbers and it is all correct. but I can't figure out how C(12,6) comes to play.
$$
\binom{12}{0} + \binom{12}... | Hint: We have
$$
\sum_{i=0}^n \binom{n}{i} = 2^n
$$
and
$$
\binom{n}{i} = \binom{n}{n-i}
$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Fibonacci sequence, proof verification. At the end of chapter $10$ of his "Book of Proof", Hammack calls for a proof of the Fibonacci sequence in a form that I did not encounter before, I would like to know if the proof that I give for it is correct.
*
*$30)$ Here $F_n$ is the $n$th Fibonacci number. Prove that... | Yes, your proof seems correct and acceptable (given the edit).
| {
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"timestamp": "2023-03-29T00:00:00",
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Show that for $p$ to be odd prime and $p \equiv 3$ mod $4$, then $x^2+y^2 = p$ has no integer solution Show that for $p$ to be odd prime and $p \equiv 3$ mod $4$, then $x^2+y^2 = p$ has no integer solution. I have no idea how can i apply quadratic reciprocity to the equation $x^2+y^2 = p$ or should use other method. ... | $p$ is odd then $x$ and $y$ have different parity. We can assume without loss of generality that $x$ is odd and $y$ is even.
We set $$x=2X+1$$
$$y=2Y$$
Then $$x^2+y^2=4X^2+4X+1+4Y^2=4(X^2+X+Y^2)+1=p$$
Hence, We only have solutions if only:
$$p\equiv 1\pmod 4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2200844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\sqrt{x^{2}+\frac{1}{n}}$ converges uniformly to $\left|x\right|$ Im trying to prove that $f_{n}\left(x\right)=\sqrt{x^{2}+\frac{1}{n}}$ converges uniformly to $f(x) = \left|x\right|$ in $[-1,1]$.
So for evary $\varepsilon$ exists $N \in \mathbb{N}$ s.t for all $n>N$ and for all $x \in [-1,1]$ $\left|f_{n}\left(x\... | We have
$$\left\vert \sqrt{x^2 + \frac{1}{n}} + \sqrt{x^2}\right\vert \geq \frac{1}{\sqrt{n}},$$
thus by your work thus far
$$\vert f_n(x) - \sqrt{x} \vert \leq \frac{\frac{1}{n}}{\frac{1}{\sqrt{n}}} = \frac{1}{\sqrt{n}}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
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If $\frac{a\sqrt{2}+b}{b\sqrt{2}+c}$ is rational, then $a+b+c|ab+bc+ca$ Let $a,b,c\in\mathbb{N_{>0}}$. Prove that if $$\frac{a\sqrt{2}+b}{b\sqrt{2}+c}$$ is a rational number, then $a+b+c|ab+bc+ca$.
Hints?
My approach was to assume that
$$ \frac{a\sqrt{2}+b}{b\sqrt{2}+c}=\frac{p}{q}$$
then
$$ pb\sqrt{2}+pc=qa\sqrt{2}+q... | Let $\frac{a\sqrt2+b}{b\sqrt2+c}=r\in\mathbb Q$.
Hence, $(a-rb)\sqrt2=rc-b$.
If $a-rb\neq0$ so we get a contradiction.
Thus, $a-rb=0$ and $rc-b=0$ or $a=rb$ and $c=\frac{b}{r}$.
Id est, $$\frac{ab+ac+bc}{a+b+c}=\frac{b^2\left(1+r+\frac{1}{r}\right)}{b\left(1+r+\frac{1}{r}\right)}=b$$
and we are done!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n} \le \frac{n}{n+1}$? I have a series $$a_n = \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n} \quad n \ge 1$$
For example, $a_3 = \frac{1}{4}+\frac{1}{5}+\frac{1}{6}$.
I need to prove that for $n \ge 1$:
$$a_n = \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n} \l... | This is wrong:
$$a_{n+1}=\frac{1}{n+1}+\frac{1}{n+3}+...+\frac{1}{2(n+1)}$$
You should have:
$$a_{n+1}=\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{2(n+1)}$$
Note that this has $1$ extra term than $a_n$ because the series is defined using $n$ twice - once for $n+1$ and again at $2n$.
From here subtract $a_n$ and rearrange ... | {
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"timestamp": "2023-03-29T00:00:00",
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How do you derive the following two summations?
I tried using the geometric series formula, $$ \sum_{n=0}^{k-1} ar^{n}=a\left (\frac{1-r^{n}}{1-r} \right ) $$, but I got $1-2\left ( \frac{1}{2}^{n-1} \right )$ for the first one and $2-2\left ( \frac{1}{2}^{n-1} \right )$ for the second.
Edit: I derived the first answ... | $\sum_{n=0}^{n-1}a\left (\frac{1-r^{n}}{1-r} \right ) =
\frac {a(n-1)}{1-r}- \frac{1}{1-r}\sum_{n=0}^{n-1} r^n $
$=\frac {a(n-1)}{1-r}- a\frac{(1-r^n)}{(1-r)^2}$ [by multiplying by $\frac{1-r}{1-r}$]
| {
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"timestamp": "2023-03-29T00:00:00",
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Blockwise cofactor matrix identity Wikipedia gives an identity for blockwise inversion, assuming the appropriate inverses exist:
$$\begin{bmatrix}\mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D}\end{bmatrix}^{-1} = \begin{bmatrix} \mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\m... | $\newcommand{\inv}[1]{{#1}^{-1}}
\newcommand{\adj}{\operatorname{adj}}
$
Let $M = \begin{pmatrix} A_{m\times m} & B_{m \times n} \\ C_{n \times m} & D_{n \times n}\end{pmatrix}$ then the following always holds
$$
\begin{pmatrix} (\det{D})^{m-1} I_m & 0 \\ 0 & (\det(A))^{n-1}I_n \end{pmatrix} \adj M = \begin{pmatrix}\ad... | {
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A natural number $a$ has four digits and $a^2$ ends with the same four digits as that of $a$. Find the value of $a$?
A natural number $a$ has four digits and $a^2$ ends with the same four digits as that of $a$. Find the value of $a$?
This question appeared in a math contest. I tried for a while and came up with this ... | The equation $a^2 = a$ has exactly two solutions ($0$ and $1$) modulo any prime power, because:
*
*Mod $p$, it has only two roots $0$ and $1$, because it's quadratic, and $\mathbb F_p$ is a field;
*By Hensel's lemma, since the derivative of $a^2-a$ is $2a-1$ and is nonzero when $a=0$ or $a=1$ we can lift those solu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2207948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Solution of non exact differential equations with integration factor depend both $x$ and $y$ I'm not finding any general description to solve a non exact equation which's integrating factor depend both on $x$ and $y$.
I'm on this problem $$(2x^{2}-y)dx+(x+y^2)dy=0 $$
I am trying to solve and kind of stuck now which is... | $(2x^2-y)~dx+(x+y^2)~dy=0$
$(y^2+x)\dfrac{dy}{dx}=y-2x^2$
Which relates to an ODE of the form http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=166.
Let $u=\dfrac{1}{x}$ ,
Then $\dfrac... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Evaluation of the series $\sum_{n=1}^{\infty} \frac{n^2+2n+3}{2^n}$. Find the following sum
$$\sum_{n=1}^{\infty} \frac{n^2+2n+3}{2^n}$$
Could someone give some hint to proceed in this question?
| The given series is absolutely convergent, and by setting $S=\sum_{n\geq 1}\frac{n^2+2n+3}{2^n}$ we also get:
$$ 2S = \sum_{n\geq 1}\frac{n^2+2n+3}{2^{n-1}} = 6+\sum_{n\geq 1}\frac{n^2+4n+6}{2^n}\tag{1}$$
from which:
$$ S = 2S-S = 6+\sum_{n\geq 1}\frac{2n+3}{2^n}\tag{2} $$
and by setting $T=\sum_{n\geq 1}\frac{2n+3}{2^... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solving $\sqrt[3]{{x\sqrt {x\sqrt[3]{{x \ldots \sqrt x }}} }} = 2$ with 100 nested radicals I have seen a book that offers to solve the following equation:
$$\underbrace {\sqrt[3]{{x\sqrt {x\sqrt[3]{{x \ldots \sqrt x }}} }}}_{{\text{100 radicals}}} = 2$$
The book also contains the answer:
$$x = {2^{\left( {\frac{{5 \ti... | Let $u_n$ be the expression with $n$ radicals. Then $u_{100} = 2$. Then $u^3_{100} = xu_{99} = 2^3$. Next $(xu_{99})^2 = x^3xu_{98} = 2^9$. Keep going and try to find a pattern.
| {
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"url": "https://math.stackexchange.com/questions/2211193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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Solve $\frac{2}{\sin x \cos x}=1+3\tan x$ Solve this trigonometric equation given that $0\leq x\leq180$
$\frac{2}{\sin x \cos x}=1+3\tan x$
My attempt,
I've tried by changing to $\frac{4}{\sin 2x}=1+3\tan x$, but it gets complicated and I'm stuck. Hope someone can help me out.
| Multiply both sides by $sin(x)cos(x)$ and rearrange to get $2-3sin^2(x)=sin(x)cos(x)$.
Square both sides: $4-12sin^2(x)+9sin^4(x)=sin^2(x)cos^2(x)$.
Substitute $1-sin^2(x)$ for $cos^2(x)$ to get $$
4-12sin^2(x)+9sin^4(x) = sin^2(x) - sin^4(x)$$
$$10sin^4(x)-13sin^2(x)+4=0$$ Substitute $u=sin^2(x)$ to get a quadratic ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Finding the general formula of the sequence $ a_n = 7a_{n-2} + 6a_{n-3} $ Given the equation $ a_n = 7a_{n-2} + 6a_{n-3} $, and $ a_0 = a_1 = a_2 = 1 $, how do I find the general equation?
I have tried to express this sequence in matrix form, $ Q = Xb $ as follows
$$ Q = \begin{bmatrix} a_{n+1} \\ a_{n-2} \\ a_{n-3} \... | We have $a_n = 7a_{n-2}+6a_{n-3}$ and we can write:
\begin{align}
a_n &= 0a_{n-1}+7a_{n-2}+6a_{n-3}\\
a_{n-1} &= 1a_{n-1} + 0a_{n-2} +0a_{n-3}\\
a_{n-2} &= 0a_{n-1}+1a_{n-2}+0a_{n-3}
\end{align}
Setting
$$A = \begin{pmatrix}
0 & 7 & 6\\
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix}\ \text{and}\ \ x_n = \begin{pmatrix} a_n\\ a_{n... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compositions of 50 with unique restrictions solve by generating function. Consider the compositions of 50 that have exactly 5 summands, such that the first and last summands are odd, all other summands are even, and no summand is greater than 20.
Use a generating function to count the number of such compositions.
Sim... | You have it all correct so far.
To alay some of your concerns I have personally always found it far easier to remember the formula for the sum of a finite geometric sequence as
$$\text{sum of geometric sequence}=\frac{(\text{first term})\: -\: (\text{term after last term})}{1\: -\: \text{ratio}}$$
In this case you have... | {
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Calculating inverse of $(7+9i)$ How can I calculate $(7+9i)^{-1}$?
So I have:
$(7+9i)^{-1}$
$(a+bi) \cdot (7+9i)$
$7a + 9ai + 7 bi + 9bi^2 = 1 + 0i$
$7a + (9a + 7b)i - 9b = 1$
So there are two equations:
1) $7a - 9b = 1$
2) $9a + 7b = 0$
So getting a from the first equation:
$a = \frac{9}{7}b$
Inserting it in the sec... | $$(7+9i)^{-1}=\frac{1}{7+9i}=\frac{1}{7+9i} \times \frac{7-9i}{7-9i}=\frac{7-9i}{7^2-(9i)^2}=\frac{7-9i}{130}= \frac{7}{130}-\frac{9i}{130}$$
Also in your solution $7a-9b=1 \implies a= \dfrac{1+9b}{7} \neq \dfrac{9b}{7}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2214667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Show $\lim_{(x,y)\to(0,0)}\frac{x^3-xy^3}{x^2+y^2}=0$ I need to show if the following limit is exists (and exists $0$),
$$
f(x,y)=\frac{x^3-xy^3}{x^2+y^2},
$$
for $\vec x\to 0$. I tried out the following:
$$
\frac{x^3-xy^3}{x^2+y^2}\leq\frac{x^3-xy^3}{x^2}=1-\frac{y^3}{x^2},
$$
but obviously I'm stuk here. A similar ap... | Note that
$$\left|x^3-xy^3\over x^2+y^2\right|\le|x|\left|x^2\over x^2+y^2\right|+|xy|\left|y^2\over x^2+y^2\right|\le|x|+|xy|$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2216579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can anything interesting be said about this fake proof? The Facebook account called BestTheorems has posted the following. Can anything of interest be said about it that a casual reader might miss?
Note that \begin{align}
\small 2 & = \frac 2{3-2} = \cfrac 2 {3-\cfrac2 {3-2}} = \cfrac 2 {3 - \cfrac 2 {3 - \cfrac 2 {3-... | $$x = \cfrac 2 {3 - \cfrac 2 {3 - \cfrac 2 {3 - \cfrac 2 {3 - \cfrac 2 {\ddots}}}}}$$
$$x = \frac 2 {3 - x}\\
x^2 - 3x + 2 = 0\\
(x-1)(x-2) = 0$$
$1,2$ are both solutions. However, if we consider this recurrence relation:
$$x_n = \frac 2 {3 - x_{n-1}}$$
when $x_{n-1} <1 \implies x_{n-1}<x_n<1$
And the squence converge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2216964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 4,
"answer_id": 1
} |
Integration by parts $\int xe^{x^2} dx$.
Find with integration by parts $$\int xe^{x^2} dx$$.
$$\int xe^{x^2} dx$$
Let $u(x) = x \ \ \ v^{'}(x) = e^{x^2}$
Now I can write the integral as $$\int xe^{x^2} dx = x \int e^{x^2} dx - \int\left(\int e^{x^2} dx \right) dx$$
Even after trying everything I am unable to solve... | Take $u=e^{x^2}$ then $du=2xe^{x^2}$ and $v=\frac{x^2}{2}$ so
$$\int xe^{x^2}dx=\frac{x^2e^{x^2}}{2}-\int x^3e^{x^2}dx$$
Take $u=e^{x^2}$ and $dv=x^3$ then
$$\int x^3e^{x^2}dx=\frac{x^4}{4}e^{x^2}-\frac{1}{2}\int x^5e^{x^2}dx$$
Similary
$$\int x^5e^{x^2}dx=\frac{x^6}{6}e^{x^2}-\frac{1}{3}\int x^7e^{x^2}dx$$
In general ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2217850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Expand binomially to prove trigonometric identity Prompt: By expanding $\left(z+\frac{1}{z}\right)^4$ show that $\cos^4\theta = \frac{1}{8}(\cos4\theta + 4\cos2\theta + 3).$
I did the expansion using binomial equation as follows
$$\begin{align*}
\left(z+\frac{1}{z}\right)^4 &= z^4 + \binom{4}{1}z^3.\frac{1}{z} + \bino... | Let $\cos \theta=z+\dfrac1z.$
Then $\cos2\theta=2\cos^2\theta-1=2z^2+3+\dfrac2{z^2}$ and similarly you can find $\cos 4\theta.$
Compare them with your binomial identity.
Good luck.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2217925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Can someone please check if my reasoning for this proof is valid or not? I have already seen the other questions about this proof. I'm just trying a different sort of method, though I'm not sure if it's valid or not.
Context for the main question: Prove by induction $2^n\gt n^3$ for $n\ge10$
Obviously, the base case wo... | Here's an alternate method: we have in our assumption that $$2^n>n^3,\quad n\geq 10$$
Then consider $f(n) = 2^{n+1}-(n+1)^3$
$f(n) = 2\cdot 2^n-n^3-3n^2-3n-1$
$f(n)= (2^n-n^3)+(2^n-3n^2-3n-1)$
Let $g(n) = n^3-3n^2-3n-1\implies g(n) <2^n-3n^2-3n-1$ using $2^n>n^3$
$g'(n) = 3n^2-6n-3 = 3(n^2-2n-1) =3[(n-1)^2-2] >0$ for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2221189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Suppose $x^2+y^2=4$ and $z^2+t^2=9$ and $xt+yz=6$. What are $x,y,z,t$? Let $x^2+y^2=4$ and $z^2+t^2=9$ and $xt+yz=6$.
What are $x,y,z,t$?
| Presumably, this should be solved over $\Bbb R$.
In this case the easiest way to think about this is that you have two (real) vectors $v = (x,y)$ and $w = (t,z)$.
Then,
$$\|v\| = \sqrt{x^2+y^2} = \sqrt 4 = 2,$$
$$\|w\| = \sqrt{t^2+z^2} = \sqrt 9 = 3,$$
$$v\cdot w = xt + yz = 6.$$
If we denote with $\varphi$ the angle b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2222245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Limit of a 2-variable function Given problem:
$$
\lim_{x\to0,y\to0}(1+x^2y^2)^{-1/(x^2+y^2)}
.$$
I tried to do it with assigning y to $y = kx$, but that didn't help me at all. Also one point, I can't use L'Hospital's rule.
| Using Polar coordinates, $x = r\cos\theta$ and $y = r\sin\theta$, the problem boils down to $$ \lim_{r\to0}\space(1 + r^4\cos^2\theta\sin^2\theta)^{\frac{-1}{r^2}}$$
Now prove that independent of the value of $\theta$, the limit exists and solve for it then.
EDIT:
Now let $$L = \lim_{r\to0}\space(1 + r^4\cos^2\theta\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2223838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $a$ , $b$, $c$ are positive integers satisfying the following $(a^2 +2)(b^2+3)(c^2+4)=2014$ What is the value of $ a+b+c $ If $a$ , $b$, $c$ are positive integers satisfying the following
$(a^2 +2)(b^2+3)(c^2+4)=2014$
What is the value of $ a+b+c $
I need details because i don't know how to solve simlar probl... | let us consider following program
z=input('enter your number : ');
string='';
for ii=2:z
s=0;
while z/ii==floor(z/ii) % check if z is divisible by ii
z=z/ii;
s=s+1;
end
if s>0
str =[num2str(ii) '^' num2str(s) ];
string=strc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2225235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Uniform convergence of $\sum_{n = 1}^\infty {\arctan{\frac{2x}{x^2+n^2}}}$ on $\mathbb R$
Check if $\sum_\limits{n = 1}^\infty{\arctan{\dfrac{2x}{x^2+n^2}}}$ is uniformly convergent series of function on $\mathbb{R}$.
"Uniform convergence" really messes up my mind.
Since $\dfrac{2x}{x^2+n^2} \approx \arctan{\dfrac{2x... | Your heuristics may give some idea, but any such heuristics should be justified in order to yield a valid proof.
Let me demonstrate a possible way. It follows from AM-GM inequality that
$$ \left|\frac{2x}{x^2 + n^2}\right| \leq \frac{1}{n} $$
uniformly in $x \in \Bbb{R}$. Together with the inequality $\frac{1}{2}|x| \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2226502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Which of the following is true for the definite integrals shown above? $$J = \int_{0}^{1}\sqrt{1-x^4}\,dx$$
$$K = \int_{0}^{1}\sqrt{1+x^4}\,dx$$
$$L = \int_{0}^{1}\sqrt{1-x^8}\,dx$$
Which of the following is true for the definite integrals shown above?
(A) $J<L<1<K$
(B) $J<L<K<1$
(C) $L<J<1<K$
(D) $L<J<K<1$
(E) $L<1<J<... | $\begin{align}L-J&=\int_0^1 \sqrt{1-x^8}dx-\int_0^1 \sqrt{1-x^4}dx\\
&=\int_0^1 \left(\sqrt{1-x^8}-\sqrt{1-x^4}\right)dx\\
&=\int_0^1 \dfrac{\left(\sqrt{1-x^8}\right)^2-\left(\sqrt{1-x^4}\right)^2}{\sqrt{1-x^8}+\sqrt{1-x^4}}dx\\
&=\int_0^1 \dfrac{x^4(1-x^4)}{\sqrt{1-x^8}+\sqrt{1-x^4}}dx>0
\end{align}$
Therefore $L>J$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2228468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Finding orthogonal surface Find the surface which is orthogonal to the system $z=cxy(x^2+y^2)$ and which passes through the hyperbola $(x^2-y^2)=a^2$.
| We can consider $c$ as function of $x,y,z$.
$G(x,y,z)=c=\dfrac{z}{xy(x^2+y^2)}$ Its gradient is a vector orthogonal to each of the surfaces $G(x,y,z)=c$ constant.
$\nabla G=-\dfrac{z(y^2+3x^2)}{yx^2(x^2+y^2)^2}\hat i-\dfrac{z(x^2+3y^2)}{y^2x(x^2+y^2)^2}\hat j+\dfrac{1}{xy(x^2+y^2)}\hat k$
We have to find a function $z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Find the number of distinct arrangements of 5 unequal positive integers such that their sum is 20 Let the integers be $n_1, n_2, n_3, n_4$ and $n_5$.
It's given in the question that $n_1 + n_2 + n_3 + n_4 + n_5 = 20$.
I thought of taking $n_2$ as $n_1 + a$, $n_3$ as $n_1 + a + b$ and so on... where a, b,... are not eq... | Integer partitions of $n$ into $k$ parts is given by the recurrence
$$p(n,k)=p(n-1,k-1)+p(n-k,k)\tag{1}\label{1}$$
Since a partition of $n$ into $k$ parts either has $1$ as it's smallest part in $p(n-1,k-1)$ ways or it has it's smallest part greater than $1$ in $p(n-k,k)$ ways. This recurrence has $p(n,1)=1$ and $p(1,1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Largest interval of definition furnished by Picard's theorem Consider the IVP $$y'=\frac{1}{(3-x^2)(9-y^2)},\;y(0)=0.$$
I need to find the largest interval of definition furnished by Picard's theorem. I know the positive endpoint of the interval is given by $\min (a,\frac bM)$ where $|x|<a,|y|<b$ and $M=\sup_B |f(x,y)|... | Picard's theorem by itself doesn't furnish a particular interval of definition, it just says there is one. There are many different ways to find an explicit interval. In this case the d.e. is separable, leading to the implicit solution
$$ - \frac{\text{arctanh}(x/\sqrt{3})}{\sqrt{3}}
- \frac{y^3}{3} + 9 y = 0 $$
Thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find coefficients of $x^{2012}$ in $(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)$ Find coefficients of $x^{2012}$ in $(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)$
Attempt: i have break $2012$ in to sum of power of $2$
as $2012 = 2^{10}+2^{9}+2^{8}+2^7+2^6+2^4+2^3+2^2$
but wan,t be able to go further, could some help me , t... | $2012 = 2047 - 35 = (\sum_{k=0}^{10}{2^k}) - 35$
$35 = 2^5+2^1+2^0 => 2012 = 2^{10}+2^9+2^8+2^7+2^6+2^4+2^3+2^2$
So this means that the for the parenthesis with the power of $x$ in this set: ${10, 9, 8, 7, 6, 4, 3, 2}$, the part with $x$ is multiplied. So for the rest of the parenthesis, the number is chosen. So the co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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transition matrix and coordinate vector Could you please help me solve these quesitons??
Consider the bases S={$u_1, u_2, u_3$}, and T={$v_1, v_2, v_3$}, with
$u_1$=[-3, 0, -3], $u_2$=[-3, 2, -1], $u_3$=[1, 6, -1],
$v_1$=[-6, -6, 0],
$v_2$=[-2, -6, 4], $v_3$=[-2, -3, 7]
(a) Find the transition matrix from $S$ to $T... | As noted by @Emilio Novati, the key idea is to connect to the standard basis.
Vectors will be colored according to the basis membership, and named chromatically:
$$
\color{blue}{\mathbf{B}\ (standard)}, \qquad
\color{red}{\mathbf{R}\ (u)}, \qquad
\color{green}{\mathbf{G}\ (v)}.
$$
$$
\mathbf{R}_{\color{red}{R}\to \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2232952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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The function $(1+x)(1+x^4)(1+x^{16})(1+x^{64})\cdots =\prod_{n\geq 0} (1+x^{4^n})$ Let $$f(x)=(1+x)(1+x^4)(1+x^{16})(1+x^{64})\cdots=\prod_{n\geq 0} (1+x^{4^n})$$
Then what is $f^{-1}(\frac{8}{5f(3/8)})?$
The answer should be a rational number.
My attempt: I tried to take a log of the expression to turn it into a sum, ... | It's hard to say things about $f(x)$ for general $x$. But we have
\begin{align}
(1-x) f(x) f(x^2) &= (1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)\dotsb \\
&= (1-x^2)(1+x^2)(1+x^4)(1+x^8)\dotsb \\
&= (1-x^4)(1+x^4)(1+x^8) \dotsb \\
&= (1-x^8)(1+x^8) \dotsb \\
&= \dots = 1.
\end{align}
(Assuming convergence.)
If $x = f^{-1}\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Find the value of $(\log_{a}b + 1)(\log_{b}c + 1)(\log_{c}a+1)$ if $\log_{b}a+\log_{c}b+\log_{a}c=13$ and $\log_{a}b+\log_{b}c+\log_{c}a=8$ Let $a,b$, and $c$ be positive real numbers such that
$$\log_{a}b + \log_{b}c + \log_{c}a = 8$$
and
$$\log_{b}a + \log_{c}b + \log_{a}c = 13.$$
What is the value of
$$(\log_{a}b... | Here's how I solved this:
First, expand the expression. For conciseness, express it as $(x+1)(y+1)(z+1)$:
$$(x+1)(y+1)(z+1) = xyz + \Big(xy + yz + xz\Big) + \Big(x + y + z\Big) + 1$$
We recognize immediately that $x+y+z$ is simply the first equation we're given, with value 8.
It must be the case that the remaining term... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2236306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
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how to calculate the multivariable limit? I need to show that the function $f(x,y)=\frac{x^5+y^6}{|x|^3+|y|^3}$ is continuous at (0,0), so I want to show that
$ \lim_{(x,y)\to(0,0)}\frac{x^5+y^6}{|x|^3+|y|^3} =0$
I tried switching to polar coordinates but did not see how that helps. what would be a good way to calcula... | It's enough to show $\;\dfrac{\lvert x^5+y^6\rvert}{\lvert x\rvert^3+\lvert y\rvert^3}$ tend to $0$. Now, setting $x=r\cos\theta$, $y=r\sin\theta$, we have
$$\dfrac{\lvert x^5+y^6\rvert}{\lvert x\rvert^3+\lvert y\rvert^3}\le\dfrac{\lvert x\rvert^5+\lvert y^6\rvert}{\lvert x\rvert^3+\lvert y\rvert^3}=\frac{r^5\bigl(\lve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to show that $\mathbb{Z}\left[\frac{1 + \sqrt{5}}{2}\right]$ is finitely generated? We can say that $\mathbb{Z}\left[\cfrac{1 + \sqrt{5}}{2}\right]$ is finitely generated if minimal polynomial of $\cfrac{1 + \sqrt{5}}{2}$ is in $\mathbb{Z}[X]$. After some calculations it can be shown that $f(X) = X^2 - X - 1 \in \... | Elements of $\mathbb Z[(1 + \sqrt{5})/2]$ take the form $g((1 + \sqrt{5})/2)$ for some polynomial $g(X) \in \mathbb Z[X]$. By the division theorem, you can write $g(X) = q(X)f(X) + r(X)$ where $r = 0$ or $\deg r < \deg f = 2$ and $q, r \in \mathbb{Z}[X]$. Then
$$ g\left( \frac{1 + \sqrt{5}}{2} \right) = q\left( \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2240560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find all $n$ for which $504$ doesn't divide $n^8-n^2$ I have seen a similar problem where someone showed that it does divide $n^9-n^3$ but here one has to show that it isn't the case for $n^8-n^2$ for all $n$.
| By the Chinese remainder theorem, $n^8-n^2$ is divisible by $504$ if and only if it is divisible by $8$, $7$ and $9$.
*
*Mod $8$, $n^2\equiv 0$ or $1$, except if $n\equiv \pm2\mod8$, hence $n^8-n^2=n^2(n^6-1)\equiv 0$, except if $n\equiv \pm2\mod8$, in which case $n^8-n^2\equiv 4$.
*Mod $7$, Lil' Fermat says $n^7\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2241697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum of all possible angles. If $$\tan\left(\frac{\pi}{12}-x\right) , \tan\left(\frac{\pi}{12}\right) , \tan \left(\frac{\pi}{12} +x\right)$$ in order are three consecutive terms of a GP then what is sum of all possible values of $x$.
I am not getting any start, can anybody provide me a hint?
| Use :$$\Big(\tan\frac{\pi}{2}\Big)^2=\tan\Big(\frac{\pi}{12}-x\Big)\tan\Big(\frac{\pi}{12}+x\Big) \tag1$$ (Condition for G.P)
And :
$$\tan (A\pm B) = \frac{\tan A \pm \tan B}{1\mp \tan A\tan B}$$
Let $\tan \dfrac{\pi}{12} =c$
Put in $(1)$
$$c^2=\frac {c+\tan x}{1-c\tan x} \cdot \frac{c-\tan x}{1+c\tan x} \implies c^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2242177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Residue of a function f
$\int \dfrac{w^2}{(w^2+1)^2(w^2+2w+2)}=\dfrac{p \pi}{q}$, limits are from minus infinity to infinity.
| After more comments see
\begin{eqnarray}
residue(f,-1+ i) &=& \dfrac{3}{25}-\dfrac{4}{25}i\\
residue(f,-1- i) &=& \dfrac{3}{25}+\dfrac{4}{25}i\\
residue(f, i) &=& -\dfrac{3}{25}+\frac{9 i}{100}\\
residue(f, -i) &=& -\dfrac{3}{25}-\frac{9 i}{100}
\end{eqnarray}
for integral $\displaystyle\int_{-\infty}^\infty \dfrac{x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2244901",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to solve this using generating functions? I have an equation:
$t_n =2t_{n-1}+t_{n-2}; t_0=0, t_1=1$
So, I rewrote this using generating functions as:
$\sum_{n=0}^{\infty} t_nz^n = 2\sum_{n=0}^{\infty} t_{n-1}z^n+\sum_{n=0}^{\infty} t_{n-2}z^n$
$g(z)=2zg(z)+z^2g(z)$
$g(z)={1\over{1-2z-z^2}}$
I don't know how to proc... | The correct and most effective approach (re. to "Concrete Mathematics") starts from writing the recurrence
in such a way that it incorporates the initial conditions and is valid for all the values of the index $n$.
Since the $t_n$ are assumed to be null for $n < 0$, we shall rewrite your recurrence as:
$$
t_{\,n} = 2\... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
} |
Detailing two integration equalities Could you please detail the follwing two equalities (I can not understand why they are valid):
\begin{align*}
\int_{0}^{\infty}\frac{\sin(2 \pi x)}{x+1}\, dx&=\int_{0}^{\infty}\frac{\sin( \pi x)}{x+2}\,dx\\
&=\int_{0}^{1}\sin( \pi x) \left[ \frac{1} {x+2} -\frac{1} {x+3}+\frac{1} {x... | The second one in the OP is not quite correct. The argument of the sine function should be $\pi x $, not $2\pi x$.
Note that we can write
$$\begin{align}
\int_0^\infty \frac{\sin(\pi x)}{x+2}\,dx&=\sum_{k=1}^\infty \int_{k-1}^k \frac{\sin(\pi x)}{x+2}\,dx\\\\
&=\sum_{k=1}^\infty \int_0^1 \frac{\sin(\pi (x+k-1))}{x+k+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.
I spent a lot of time trying to solve this and, having consulted some ... | This is not a proof in itself, but if you've studied statistics, then you've seen a proof that
$$0\le V(X)=E(X^2)-E(X)^2$$
If we now consider a random variable $X$ with three equally likely values, $X=x,y$, and $z$, then we have
$$E(X)={x+y+z\over3}\qquad\text{and}\qquad E(X^2)={x^2+y^2+z^2\over3}$$
If, in addition, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2247973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 15,
"answer_id": 7
} |
Prove that $\lim_{(x,y)\to (0.0)} \frac{4xy^2 - 3x^3}{x^2 - y^2}$ does not exist I'm struggling with this limit. I have to approach using different curves and show that there is one curve wich prove that this limit does not exist, despite the fact that when trying with a lot of curves shows that the limit is 0.
If some... | By writing $x,y$ in polar coordinates,
$$\lim_{(x,y)\to (0,0)} \frac{4xy^2 - 3x^3}{x^2 - y^2}=\lim_{r\to 0}\ r\cdot\frac{4\cos\theta\sin^2\theta - 3\cos^3\theta}{\cos^2\theta - \sin^2\theta}$$
indicating that the limit may not be zero only when $\tan^2\theta\to 1$, that is $\theta\to \frac{\pi}{4}$ or $\theta\to\frac{7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2249619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$2x^2 + 3x +4$is not divisible by $5$ I tried by $x^2 \equiv 0, 1, 4 \pmod 5$ but how can I deal with $3x$?
I feel this method does not work here.
| $5$ is small. You could just consider all possibilities $\mod{5}$. There are only $0,1,2,3,4$. If any of these get you $0$, then, it can be possibly divisible by $5$. Otherwise, not. Let $f(x)=2x^2+3x+4$. So, $x\equiv0\implies f(0)\equiv4$, $x\equiv1\implies f(1)\equiv 2+3+4\equiv4\pmod{5}$, $x\equiv2\implies f(2)\equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2250718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 2
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If $ab$ is a square number and $\gcd(a,b) = 1$, then $a$ and $b$ are square numbers.
Let $n, a$ and $b$ be positive integers such that $ab = n^2$. If $\gcd(a, b) = 1$, prove that there exist positive integers $c$ and $d$ such that $a = c^2$ and $b = d^2$
So far I have tried this:
Since $n^2 = ab$ we have that $n = \s... | This:
$\sqrt{a}$ and $\sqrt{b}$ are both positive integers
does not follow from
$\sqrt{a}(k\sqrt{a}) + \sqrt{b}(l\sqrt{b}) = 1$.
For instance,
$$ \sqrt{2}((-1)\sqrt{2}) + \sqrt{3}((1)\sqrt{3}) = 1$$
($a = 2, b = 3, k = -1, l = 1$).
What you need to do instead is use prime factorization.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2253940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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Find $\lim\limits_{x \to 0}{\frac{1}{x}(\frac{1}{\sin(x)}-\frac{1}{\sinh(x)})}$.
$$\lim\limits_{x \to 0}{\frac{1}{x}\left(\frac{1}{\sin(x)}-\frac{1}{\sinh(x)}\right)}$$
I replaced $\sinh(x)$ by $\displaystyle\frac{e^{x}-e^{-x}}{2}$ and used L'Hopital's rule twice but this expression becomes very large.
Could you ple... | Disclaimer: I am going to write the mother of overkills, just for fun.
From the Weierstrass product for the sine function we have
$$ \frac{x}{\sin(x)} = \prod_{n\geq 1}\left(1-\frac{x^2}{\pi^2 n^2}\right)^{-1} = \prod_{n\geq 1}\left(1+\frac{x^2}{\pi^2 n^2}+\frac{x^4}{\pi^4 n^4}+\ldots\right) \tag{1}$$
and since $\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2255162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
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How can all the options be correct in this question? Question: The equations of two sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively. If the area of the Triangle $ABC$ is 5 sq. units , then the possible equations of the side $BC$ is (are): ?
$A). x-3y+1=0$
$B). 3x+y+2=0$
$C). x-3... | WLOG, $B(a,5-a);C(b,7b-3)$
We have $$(a-1)^2+(5-a-4)^2=(b-1)^2+(7b-3-4)^2$$
$$\iff2(a-1)^2=50(b-1)^2\implies a-1=\pm5(b-1)$$
For $a-1=5(b-1), a=5b-4$
$\implies|BC|^2=(a-b)^2+(5-a-7b+3)^2=(5b-4-b)^2+\{8-7b-(5b-4)\}^2=(b-1)^2(4^2+12^2)$
Now if $M$ is the midpoint of $BC,M:\left(\dfrac{a+b}2,\dfrac{5-a+7b-3}2\right)$
Repl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Summation splitting: $\sum_{n=1}^{\infty} \frac{n}{3\cdot 5\cdot 7 \dots (2n+1)}$ $$\sum_{n=1}^{\infty} \frac{n}{3\cdot 5\cdot 7 \dots (2n+1)}$$
Can someone please help me in solving this problem, I tried to take the summation of the numerator and denominator individually but my teacher said that it is wrong to do the ... | Hint. One may observe that, for $n\ge1$,
$$
\begin{align}
\frac {n}{1\cdot 3\cdots (2n+1)} &=\color{red}{\frac12} \cdot \frac {(2n+1)-1}{1\cdot 3\cdots (2n+1)}
\\\\&=\color{red}{\frac12} \cdot \frac {1}{1\cdot 3\cdots (2n-1)}-\color{red}{\frac12} \cdot \frac {1}{1\cdot 3\cdots (2n+1)},
\end{align}
$$ then, by telescop... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Write the function $\frac{1}{(z+1)(3-z)}$ as a Laurent series. $$f(z)=\frac{1}{(z+1)(3-z)}=\frac{1}{4z+4} + \frac{1}{12-4z}$$
$$\frac{1}{4z+4}=\frac{1}{4z}\frac{1}{1-\frac{-1}{z}}=\frac{1}{4z}\sum_{k=0}^{\infty} \left(\frac{-1}{z}\right)^k$$
$$\frac{1}{12-4z}=\frac{1}{12}\frac{1}{1-\frac{z}{3}}=\frac{1}{12}\sum_{k=0}^{... |
The function
\begin{align*}
f(z)&=\frac{1}{(z+1)(3-z)}\\
&=\frac{1}{4(z+1)}-\frac{1}{4(z-3)}
\end{align*}
has two simple poles at $-1$ and $3$.
Since we want to find a Laurent expansion with center $0$, we look at the poles $-1$ and $3$ and see they determine three regions.
\begin{align*}
|z|<1,\qquad\quad
1<|z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2257748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving that $3^n+7^n+2$ is divisible by 12 for all $n\in\mathbb{N}$. Can someone help me prove this? :( I have tried it multiple times but still cannot get to the answer.
Prove by mathematical induction for $n$ an element of all positive integers that $3^n+7^n+2$ is divisible by 12.
| With induction:
When $n=1$: $3^1+7^1+2=12$ is divisible by $12$. Therefore, we assume that $(3^k+7^k+2)$ is divisible by $12$ for some $k≥1$ (thus, $3^k+7^k+2=12a$ where $a$ is a function of $3$ and $7$).
Then, $$3^{k+1}+7^{k+1}+2=3×3^k+7×7^k+2$$
$$=3×3^k+7^k(3+4)+6-6+2$$
$$=3×3^k+3×7^k+6+4×7^k-4$$
$$=3(3^k+7^k+2)+4(7^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2258457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Which is greater $x_1$ or $x_2$? $$x_1=\arccos\left(\frac{3}{5}\right)+\arccos\left(\frac{2\sqrt{2}}{3}\right)$$
$$x_2=\arcsin\left(\frac{3}{5}\right)+\arcsin\left(\frac{2\sqrt{2}}{3}\right)$$ We have to find which is greater among $x_1$ and $x_2$
If we add both we get $$x_1+x_2=\pi$$ If we use formulas we get
$$x_1=\a... | Ok fine similarly we have $$x_1=arccos\left(\frac{3}{5}\right)+arccos\left(\frac{2\sqrt{2}}{3}\right)=arcsin\left(\frac{4}{5}\right)+arcsin\left(\frac{1}{3}\right)$$
Now $$\frac{4}{5} \lt \frac{\sqrt{3}}{2}$$
hence
$$arcsin\left(\frac{4}{5}\right) \lt \frac{\pi}{3}$$ and
$$\frac{1}{3} \lt \frac{1}{2}$$ so
$$arcsin\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2259157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
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Proving that a sequence is monotone Let $ \{s_m\}_{m=1}^\infty$ be the sequence defined:
$s_m = \sum_{k=1}^{m} \frac{1}{\sqrt{m^2 + k}}$
I have already proven that $s_m\to 1$ as $m\to \infty $ but i'm having trouble to show that $s_m$ is a monotonic sequence.
Would appreciate any help or advice, thanks in advance.
| The power series for
$(1+x)^{-1/2}$
with $0 < x < 1$
is an enveloping series,
in that its sum is between
any two consecutive finite sums.
Since
$(1+x)^{-1/2}
=1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+...
$
we have
$1-\frac{x}{2}
\lt (1+x)^{-1/2}
\lt 1-\frac{x}{2}+\frac{3x^2}{8}
$
or
$(1+x)^{-1/2}
=1-\frac{x}{2}+ax^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2259627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Determining the number of solutions using Generating functions I have the equation
$u_1 + u_2 + ... + u_5 = 24$
with the restrictions
$1 \le u_i \le 7, i = 1,...5$
I've managed to work up to the point of finding the coefficient of $x^{24}$ in
$(X+X^2+...X^7)^5 = X^5(1+X+...X^6)^5$
I know my final answer will need to be... | Everything is fine with your generating function. It is convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ in a series.
We obtain
\begin{align*}
[x^{24}]&(x+x^2+\cdots+x^7)^5\\
&=[x^{24}]x^5(1+x+\cdots+x^6)^5\\
&=[x^{19}]\left(\frac{1-x^{7}}{1-x}\right)^5\tag{1}\\
&=[x^{19}](1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Laurent series of $\frac{1}{z^4+z^2}$ about $z=0$ Okay, I get $\frac{1}{z^4+z^2}=\frac{1}{z^2}-\frac{1}{z^2+1}$
And I know I need to use geometric series sum but couldn't quite find it. Can someone give me a hint?
| One of the two terms is easy $$\frac{1}{1+z^2} = \frac{1}{1-(-z^2)}=\sum_{m=0}^{+\infty} (-1)^mz^{2m}.$$ For the other, notice that $\frac{1}{z^2}$ is already in a Laurent form at $z=0.$ So the developpement is $$\frac{1}{z^4+z^2} = \frac{1}{z^2} - \sum_{m=0}^{+\infty} (-1)^mz^{2m}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Volume of the revolution solid Let $K \subset \mathbb R^2$ be area bounded by curves $x=2$, $y=3$, $xy=2$ and $xy=4$.
What is the volume of the solid we get after rotating $K$ around $y$-axis?
I tried to express the curves in terms of $y$ and solved the integral $\displaystyle \pi\left(\int_2^3 \frac{16}{y^2}dy - \int_... | $y = 3$ intersects $xy = 2$ at $x = \frac 23$ and $xy = 4$ at $x = \frac 43$
By shells I get.
$2\pi\int_\frac {2}{3}^{\frac 43} (3-\frac 2x)x \ dx + 2\pi\int_{\frac 43}^2 (\frac 4x -\frac 2x)x \ dx\\
2\pi\int_\frac {2}{3}^{\frac 43} 3x-2 \ dx + 2\pi\int_{\frac 43}^2 2\ dx\\
2\pi(\frac{3}{2}x^2 -2x|_\frac {2}{3}^{\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2262917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Monotonicity of $f(x,y)=\frac{x^2+y^2}{x+y}$ $f(x,y)=\frac{x^2+y^2}{x+y} $ where $x,y \in [0,1]$
let $x\leq r$ and $y\leq s$ I have to prove\disprove $f(x,y)\leq f(r,s)$.
Clearly $x^2+y^2\leq r^2+s^2$ and $x+y\leq r+s$ but then $\frac{1}{x+y}\geq \frac{1}{r+s}$ ??
| It is not true.
As one of many counterexamples, consider $x=r=1$ and $y=0, s=\frac12$
Then $\dfrac{1^2+0^2}{1+0}=1 \gt \dfrac56 = \dfrac{1^2+\left(\frac12\right)^2}{1+\frac12}$ so $f(x,y) \not \le f(r,s)$ even though $x \le r$ and $y \le s$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2263952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Is it true for any associative groupoid which elements are $a,b,c$, that $(a\ \circ \ b) \circ c = (b \ \circ c) \circ a $? As I know an associative groupoid is a semigroup. I'm assuming that it can be true, but I'm not sure, how can I give a proof for it?
| Let $G = (\{a,b,c\},\circ)$ be the groupoid where $\circ$ obeys the following table:
$$
\begin{array}{c|ccc} \circ & a & b & c \\ \hline a & a & a & a \\ b & b & b & b \\ c & a & a & c \end{array}
$$
One can check that $\circ$ is an associative operation. However
$$(a\circ b)\circ c = a \circ c = a \neq b = b\circ a = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2264413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability problem with tissues Girl A has 10 new tissues and 5 used tissues in her drawer. Girl B comes to visit girl A. Girl A makes her friend a coffee and randomly takes out 2 tissues out of the drawer. Later she puts them back into the drawer. The next day she takes out 2 tissues again. What is the probability th... | Initially we have ten unused tissues and five used ones. There are three different use cases:
*
*The girl takes out two used tissues on the first day. Both the first and the second drawn tissue must be used, so this probability equals $\frac{5}{15} \cdot \frac{4}{14}$;
*The girl takes out one used and one unused ti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2265263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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How many four digit numbers divisible by 29 have the sum of their digits 29?
How many four digit numbers divisible by $29$ have the sum of their digits $29$?
A way to do it would be to write $1000a+100b+10c+d=29m$ and $a+b+c+d=29$ and then form equations like $14a+13b+10c+d=29m'$ and eventually $4a + 3b – 9d = 29 (m'... | $(a=4\land b=9\land c=8\land d=8\land m=172)\lor (a=7\land b=5\land c=9\land
d=8\land m=262)\lor (a=7\land b=8\land c=5\land d=9\land m=271)\lor (a=9\land
b=6\land c=8\land d=6\land m=334)\lor (a=9\land b=9\land c=4\land d=7\land
m=343)$
So 5 numbers which are $4988,7598,7859,9686,9947$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2266567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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A number that is a triangle number and also a square number. This is a problem from standupmaths, from Youtube.
Question:
Think of a number which is a triangle number and a square number.
This can be expressed using a single equation:
$$x^2=\frac{y(y+1)}{2}$$
If you use trial and improvement, you have the square num... | Following Shark, here is how to solve the Pell equation $z^2 - 8 x^2 = 1$ by hand, although one can easily guess the first as $9-8=1:$
Method described by Prof. Lubin at Continued fraction of $\sqrt{67} - 4$
$$ \sqrt { 8} = 2 + \frac{ \sqrt {8} - 2 }{ 1 } $$
$$ \frac{ 1 }{ \sqrt {8} - 2 } = \frac{ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2267316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$ If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$
My Attempt:
$$\sin x + \sin^2 x=1$$
$$\sin x = 1-\sin^2 x$$
$$\sin x = \cos^2 x$$
Now,
$$\cos^8 x + 2\cos^6 x + \cos^4 x$$
$$=\sin^4 x + 2\sin^3 x +\sin^... | Here is a shorter way to do it. Your condition can be rewritten in the forms
$$\sin x = \cos^2(x) \implies \sin^n(x) = \cos^{2n}(x) \implies \sin^n(x)+\sin^{n-1}(x) = \sin^{n-2}(x)$$
We thus rewrite your cosines as
$$\begin{align}
\cos^8 x + 2\cos^6 x + \cos^4 x &= \sin^4(x)+2\sin^3(x)+\sin^2(x)\\ &= [\sin^4(x)+\sin^3(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2269298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
} |
For any value of $n\ge5$, the value of $1+ \frac{1}{2}+ \frac{1}{3}+...+\frac{1}{2^n+1}$ lies between
For any value of $n\ge5$, the value of $$1+ \frac{1}{2}+ \frac{1}{3}+...+\frac{1}{2^n+1}$$ lies between
A) $0$ and $\frac{n}{2}$;
B) $\frac{n}{2}$ and $n$;
C) $n$ and $2n$;
D) none of them
I know this is Harmonic pro... | Note that
$$\sum\limits_{k=1}^{2^n+1} \dfrac{1}{k} > \sum\limits_{k=2}^{2^n}\dfrac{1}{k} = \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + \dots \dfrac{1}{2^n} > \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{8} + \dots + \dfrac{1}{2^n} = \sum\limits_{k=1}^n \dfrac{2^{k-1}}{2^k} = \dfrac{n}{2}$$
On ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2272741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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extrema of functions of two variables An experiment was conducted by a group of students to analyze the performance of a subject if stimulus A and stimulus B are used. It was found that if $x$ units of stimulus A and $y$ units of stimulus B were applied, the performance of the subject can be measured using the followi... | Since you haven't mentioned it, I am assuming that stimuli A and B both take values in ${\rm I\!R}$.
Let's compute the gradient of the function we want to maximize:
$$
\nabla f (x, y) = \begin{pmatrix}
\frac{\partial f}{\partial x} (x, y)
\\
\frac{\partial f}{\partial y} (x, y)
\end{pmatrix}
= \begin{pmatrix}
(1-2x^2)y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2272947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Calculating general $n\times n$ determinant I'm given determinant $\begin{vmatrix}
1 & 2 &3 & \cdots & n -1 & n \\
2 & 3 &4 & \cdots & n & 1 \\
3 & 4 &5 & \cdots & 1 & 2 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
n & 1 & 2 & \cdots & n-2 & n-1
\end{vmatrix}$ and I have to calculate it's value only us... | In your second step, when you subtract row $i-1$ from row $i$, there should be a diagonal of $n$'s under your diagonal of $-n$'s, because $1-(1-n)=n$. For example, the third entry in the last column of your third determinant should be $n$, not $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2273869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find all natural numbers $n$ such that $3^n+81$ is a perfect square. Find all natural numbers $n$ such that $3^n+81$ is a perfect square.
My solution:
$3^n=a^2-81 \Rightarrow 3^n=(a-9)(a+9)$
$gcd(a-9,a+9) \mid 18$
because $a+9>a-9$ then $a-9$ can be $1,3,9$that only $a=9$ is the answer so we have only one solution.I ju... | You get $3^n = (a-9)(a+9)$ in the question. But the only factors of $3^n$ are other powers of $3$, so you need $a-9$ and $a+9$ to both be powers of $3$. The only powers of three differing by $18$ are $9$ and $27$. So $a = 18$, and the only solution is $18^2 = 3^5 + 81$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Given that $x+\frac{1}{x}=\sqrt{3}$, find $x^{18}+x^{24}$ Given that $x+\frac{1}{x}=\sqrt{3}$, find $x^{18}+x^{24}$
Hints are appreciated. Thanks in advance.
| As $x\ne0,$ on simplification we find $$x^2+1=\sqrt3x$$
Squaring we get $$x^4+2x^2+1=3x^2\iff x^4-x^2+1=0$$
Now
$$ x^6+1=(x^2+1)(x^4-x^2+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2276746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
For integer m greather than 2, $\frac{1}{m} + \frac{1}{m+2}$, the numerators and denomitors are primitive pythagorean triples $a$ and $b$ For $m = 2$, the fraction is $\frac{3}{4}$. for $m=3$, the fraction is$\frac{8}{15}$. I was wondering why numerators and demoninators of $\frac{1}{m} + \frac{1}{m+2}$ show primitive... | There is a simple form to proof:
$(m^2+2m)^2 + (2(m+1))^2 = ((m+1)^2 - 1)^2 + 4 (m+1)^2 = (m+1)^4 + 1 -2(m+1)^2 + 4(m+1)^2 = (m+1)^4 -2(m+1)^2 + 1 = ((m+1)^2 - 1)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2277117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Proving Ramanujan's Nested Cube Root Ramanujan's Nested Cube:
If $\alpha,\beta$ and $\gamma$ are the roots of the cubic equation$$x^3-ax^2+bx-1=0\tag{1}$$then, they satisfy$$\alpha^{1/3}+\beta^{1/3}+\gamma^{1/3}=(a+6+3t)^{1/3}\tag{2.1}$$
$$(\alpha\beta)^{1/3}+(\beta\gamma)^{1/3}+(\gamma\alpha)^{1/3}=(b+6+3t)^{1/3}\tag{... | Let $$\sqrt[3]{\cos\tfrac {2\pi}7}+\sqrt[3]{\cos\tfrac {4\pi}7}+\sqrt[3]{\cos\tfrac {8\pi}7}=x$$ and
$$\sqrt[3]{\cos\tfrac {2\pi}7\cos\tfrac {4\pi}7}+\sqrt[3]{\cos\tfrac {2\pi}7\cos\tfrac {8\pi}7}+\sqrt[3]{\cos\tfrac {4\pi}7\cos\tfrac {8\pi}7}=y.$$
Hence, since $$\cos\tfrac {2\pi}7+\cos\tfrac {4\pi}7+\cos\tfrac {8\pi}7... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Binomial coefficients equality or maybe probability Let $m,n$ be positive integers.
Evaluate the following expression:
$$
F(m,n) = \sum\limits_{i=0}^n\frac{\binom{m+i}{i}}{2^{m+i+1}}+
\sum\limits_{i=0}^m\frac{\binom{n+i}{i}}{2^{n+i+1}}.
$$
Calclulations give the hypothesis that $$F(m,n)=1,$$ for all positive integers ... | Here is an answer in two steps based upon generating functions
First step: The following identity holds true for $m,n\geq 0$
\begin{align*}
\sum_{i=0}^n\binom{m+i}{i}\frac{1}{2^{m+i+1}}=\frac{1}{2^{m+n+1}}\sum_{i=0}^n\binom{m+n+1}{i}\tag{1}
\end{align*}
In order to show (1) it is convenient to use the coefficient o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2278964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to calculate the volume of an ellipsoid with triple integral I'm having some troubles since this morning on an exercise.
I need to find the volume of the ellipsoid defined by $\frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{a^2} \leq 1$.
So at the beginning I wrote
$\left\{\begin{matrix} -a\leq x\leq a
\\ -b\leq y... | If you are sticking with cartesian coordinates, then you will consider the points inside of the ellipsoid. The limits you wrote in your OP consider all points within $\pm a$,$\pm b$,$\pm c$ as @u8y7451 mentioned.
First, let's find our range of z.
$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \le 1$
$\frac{z^2}{c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2279468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"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.