Tagged Questions

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Do I need to use induction to have sufficient rigor in this proof?

I'm taking my first analysis class this summer. The professor asked us to prove that $a^{2n}-b^{2n}$ is divisible by $a+b$. After dorking around with the first couple of $n$ I was able to come up ...
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Find count of all combination of numbers whose sum is x

I want to find the sum of all combination of numbers whose sum is x, for e.g. when x = 3 f(x) = countOf(111,12,21,3) = 4
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Recursive sequence problem

$$U(n+1) = (6+U(n))^{1/3},\text{ and } U(0) = 1.$$ Prove by induction that for all positive integers $n, U(n)$ is increasing. Prove by induction that for all positive integers $n, U(n) \leq 2$ ...
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How to prove that for any $n$ in $\mathbb{N}$ that $(\frac{3}{2})^n \ge n$?

Well, I was trying to do that using proof by induction and my attempt is : Base case : $(\frac{3}{2})^0 \ge 0$, true Assumption : $(\frac{3}{2})^k \ge k$. I've multiplied both sides by $(\frac{3}{2})$ ...
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Proving the AM:GM inequality

I am doing past exam papers preparing for the finals and I came across this questions about three times: Prove that: $$\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\geq \sqrt[n]{a_{1}.a_{2}...a_{n}}$$ ...
I have this relation $u_{n+1}=\frac{1}{3}u_{n} + 4$ and I need to express the general term $u_{n}$ in terms of $n$ and $u_{0}$. With partial sums I found this relation $u_{n}=\frac{1}{3^n}u_{0} + ... 10answers 1k views Prove by mathematical induction that$1 + 1/4 +\ldots + 1/4^n \to 4/3\$
Please help. I haven't found any text on how to prove by induction this sort of problem: $$\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}$$ I can't ...