How to prove $\sum_{k\leq n}^{n} \binom{n}{k}= 2^n$ by induction
88628
Proving the total number of subsets of S is equal to $2^n$
108596
Proof by induction for "sum-of"
32791
Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$
10777
Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
79652
Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$
23557
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
125220
Solving binomial theorem via induction
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
27922
prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
67527
Proof by Induction that $3^n ≥ 1+2^n$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
120074
Prove by Combinatorial Argument that $\binom{n}{k}= \frac{n}{k} \binom{n-1}{k-1}$
35342
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
44161
Using induction to prove $\sum_{k=0}^{N-1} 2^k = 2^N-1$
13525
Prove: If $n=2^k-1$, then $\binom{n}{i}$ is odd for $0\leq i\leq n$
18046
Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $