Prove that $\sum_{i=0}^{n-1} {2^i} = 2^n -1$
108318
counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
50226
Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
20317
Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
151646
Prove by induction that $\sum_{i=1}^{n} i \cdot 2^i = (n-1) \cdot 2^{n+1} +2$. Help finding my mistake
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
150642
How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
15164
Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
36639
Prove by induction that $\sum_{r=0}^{n}\begin{pmatrix}n \\ r \end{pmatrix}=2^k$
108298
Prove $\sum_{i=1}^{i=n}\frac{a_i}{S-a_i}\geqslant \frac{n}{n-1}$ and $\sum_{i=1}^{i=n}\frac{S-a_i}{a_i}\geqslant n(n-1)$.
99052
Prove that $( 1 + n^{-2}) ^n \to 1$.
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
40784
$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
8360
$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
142666
Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$
32791
Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$