jojobo
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If n is divisible by 4 and 2 then why it is not divisible by 8?
3 votes

In general your rule holds if $n$ and $m$ are coprime. Otherwise the common factor may be only one time in the prime factorisation, but is two times in the product. In your case: $2$ and $4$ are both ...

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$(1-t^2)^2 = 1-2t^2 + t^4$ . Why $+ t^4$?
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2 votes

By the binomial formula we have $(1-t^2)^2=1-2 \times t^2+{(t^2)}^2=1-2t^2+t^{2 \times 2}=1-2t^2+t^4$.

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Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
2 votes

It is equivalent to $\nexists n: \frac{n^2+1}{p}\in \mathbb{N}$, which is a case of the sum of squares theorem.

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Name (or proof) of $\frac{a}{b} - 1 = \frac{a-b}{b}$
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2 votes

Its just $\frac{a}{b}-1=\frac{a}{b}-\frac{b}{b}=\frac{a-b}{b}$ as @Randall mentioned.

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How to prove this logarithmic function?
2 votes

By the rules for logarithms we have $-\log(\frac{\sqrt{1+x^2}-1}{x})=\log(\frac{x}{\sqrt{1+x^2}-1})=\log((\frac{x}{\sqrt{1+x^2}-1}^2)^{\frac{1}{2}})=\frac{1}{2}\log(\frac{x^2}{(\sqrt{1+x^2}-1)^2})$. ...

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How to simplify this logarithm to find the value of x?
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2 votes

By some easy algebraic computations, we get $x^{\log_2x}=2^{\log_2(x^{\log_2x})}=2^{\log_2x \times \log_2x}=2^{\log_2^2x}$. $\Rightarrow \log_2^2x = 4$ $\Rightarrow \log_2x = \pm 2$ $\Rightarrow ...

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Prime Sequence Conjecture
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2 votes

As @DanielFischer pointed out, the conjecture is true: $a=b-2$ $\Rightarrow c=(a+b)+(a*b)$ $=(b-2+b)+((b-2)*b)$ $=b^2-2$. Thus, $c+2=b^2$ is always composite.

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determine the convergence of this infinity serie
1 votes

To avoid confusion, I will call your $p$ in my answer $r$. The series $\sum_{n=1}^{\infty}n^{-p}$ is called $p$-series and known to be convergent iff $p>1$. Thus, your series is convergent if and ...

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$f$ such that $\frac{an}{n-b} \leq f(a,b)\cdot n $ for all $n \geq 1$
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1 votes

Assuming that $n$ is a natural number, you should divide the inequality by $n$ as @RossMillikan mentioned in the comments. Then you get $\frac{a}{n-b} \leq f(a,b)$ for all $n \geq 1$. Now it is easy ...

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Show that sequence $a_n$ defined by $a_{n+1} := \frac{(a_n + b_n )}{2}$, $b_{n+1} := \sqrt{a_n b_n}$ is decreasing
1 votes

The case $a_1=b_1$ is trivial, so we assume $a_1>b_1$. Then by the AM-GM inequality we get $b_{n+1}<a_{n+1}$ and $a_{n+2}=\frac{a_{n+1}+b_{n+1}}{2}<\frac{a_{n+1}+a_{n+1}}{2}=a_{n+1}$.

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$\lim\limits_{n\to \infty}\frac{3^n+6^n-1}{6^n-1}$
1 votes

$\frac{2^{-n}}{1-6^{-n}}<\frac{1}{50}$ is equivalent to $2^n-3^{-n}>50$. By $3^{-n}<1$ if $n>1$ you get the solution.

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How solve an Increase function problem
1 votes

$\frac{f(12)-f(10)}{12-10}=\frac{156-110}{2}=\frac{46}{2}=23$

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Find % difference between a positive and a negative number
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1 votes

The percentage difference is defined as the difference of the values, divided by the average and written in percentage. If you have two values, the formula is $\frac{|x-y|}{\frac{x+y}{2}} \times 100$. ...

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testing if the infinite series are converges or diverges
1 votes

As @Yuval showed, the first one diverges. Since the integral test is not allowed, you can use direct comparison test: Comparing your series to the harmonic series $\sum_{k=1}^{\infty}\frac{1}{k}$, ...

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How do I turn this into a sigma function?
Accepted answer
1 votes

I think the following works: $$\sum_{k=2}^{\lfloor\frac{n}{3}\rfloor} \sum_{j=1}^{min(k-1,n-3k-1)} \lceil \frac{n-(3 k+j)}{2} \rceil$$ Explanation The outer sum follows from your $3 \times k$, ...

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Partition 100 people, 4 from each country into 4 groups with conditions
1 votes

This questions seems to be very similar to graph theory problems and as @MisraLavrov has showen, its solvable with methods from graph theory. One example of those methods is the concept of vertex-...

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Proof by induction: Inductive step struggles
0 votes

\begin{align} \frac{2^{k+1}+(-1)^k}{3 \times 2^k}+\left(-\frac{1}{2}\right)^{k+1} &=\frac{2 \times 2^{k+1}+2 \times(-1)^k}{3 \times 2^{k+1}}+\frac{3 \times (-1)^{k+1}}{3 \times 2^{k+1}} \\ &=\...

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Showing that a 2-variable function is decreasing with respect to a given variable on an interval
0 votes

$F(x,y)=-\frac{\log\left( \frac{x+3}{x}\right)+\frac{1}{2}\log\left(\frac{x^2}{x^2-1} \right)+\frac{1-4y}{2}\log\left( \frac{x+1}{x-1} \right)-2\sqrt{\frac{x+3}{x}}+3-4y}{-\log\left( \frac{x+3}{x}\...

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Divisors of n! including 1 which sum to n!
0 votes

Write $n!=2 \times \prod_{k=2}^{n-1}(1+k)$, then expand the product to get a subset of the divisors of $\frac{1}{2}n!$. Now you just need to multiply each term by $2$. Since you want one summand to be ...

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What are the math processes to get from $1-a$ to $1 - \frac{.1a}{1-.9a}$?
0 votes

Obviously the case $a=0$ holds. Since $$1-a=1-\frac{0.1a}{1-0.9a} \Rightarrow 1=\frac{0.1}{1-0.9a} \Rightarrow 1-0.9a=0.1a \Rightarrow 1=a,$$ your equations hold iff $a \in \{0,1\}$.

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How to convert 3 numbers (or axes) into one number
0 votes

A very natural solution is to weight the values. For example take $f(x)=1 \times$ positive $+ 0 \times $ neutral $ + (-1) \times $ negative. It's easy to show that $f(x) \in [-1,1]$ and why you can ...

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Assume $f: \mathbb R \to \mathbb R$ is continuous on $\mathbb R$ and $\lim_{x \to +\infty} f(x)=0=\lim_{x \to -\infty} f(x)$
0 votes

Sometimes there is only one of max and min. Take for example $f(x)=\frac{1}{x^2+1}$, which answers your question about "and" / "or". Also your argumentation does not work: Every ...

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How do I solve the following recursive equation?
0 votes

Hint: Write it as a product and use $1+\frac{1}{101-n}=\frac{102-n}{101-n}$, then you can apply the telescoping technique.

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Is it possible to write this rational sequence : $(u_n=\frac{2^{n-1}+1}{2^n})_{n \ge 1}$
0 votes

As mentioned in the comments, you can use a recurrence, which needs only $\frac{1}{2},+$ and $\times$: Since $u_n=\frac{1}{2}+\frac{1}{2^n} \Rightarrow \frac{1}{2^n}=u_n-\frac{1}{2}$, we have $u_{n+1}...

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how can calculate discrete value of $sinc^2x$
0 votes

In the following links there are many information about the function $sinc(x)$ and it's values: Definition of Sinc function (another question with complete answers) https://en.m.wikipedia.org/wiki/...

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How to prove this identity: $4\cdot(\frac12)!=\pi$
0 votes

Not a answer, but important to know: As Qiaochu Yuan mentioned, the Gamma function gives your equality. But in the usual definition of the factorial it's undefined. That means the Gamma function is ...

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Polynomial equation root
0 votes

The argumentation in you're answer seems to be correct, although I haven't calculated it all. You should add the possibility, that P only "touches" the x-axis in $x^*$. Consider for example ...

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