Proof by induction $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n\cdot 2^{n-1}$ for each natural number $n$
54442
Sum of $k {n \choose k}$ is $n2^{n-1}$
29828
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
33082
Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.
9522
Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
111298
Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
725
proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$
13291
Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction
115459
Proving this $F_{n+1} \cdot F_{n-1} - F^2_n = (-1)^n$ by induction
44328
Prove that $\sum_{k=1} ^ n k^{-1} = \ln(n) + O(1)$
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
55481
Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$
101899
Proof By Induction of a Sum
26901
induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
132486
Induction proof on Fibonacci sequence: $F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$
155769
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
44161
Using induction to prove $\sum_{k=0}^{N-1} 2^k = 2^N-1$