Induction: $\sum_{k\mathop=1}^n k = \frac{n(n+1)}2$ and $\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2$
16812
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
16177
Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
11612
How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
30518
Formula for $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$?
121544
Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
115912
Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$
143113
Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
118018
Show $ (n+1)^{1/3}-n^{1/3}=O\left(n^{-2/3}\right) $
114280
Computing $\sum_{n=1}^∞\frac{1}{(n+1)(n+2)(n+3)....(n+p)}$
136737
Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$
36639
Prove by induction that $\sum_{r=0}^{n}\begin{pmatrix}n \\ r \end{pmatrix}=2^k$
136274
$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
20271
Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
107824
Proof by induction that $a_0 = 1, a_1 = 1, a_n=2a_{n-1} + 3a_{n-2}$ satisfies $a_n = \frac12 (3^n) + \frac12 (-1)^n$
13291
Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction
123427
Proving the sum of the first $n$ natural numbers by induction
96332
Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$