Proof that $C(n , 0) + C(n+1 , 1) + C(n+2 , 2) + ... + C(n + k , k) =C(n+k+1 , k)$
8476
Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
111298
Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
71564
Combinatorial proof of identity involving central binomial coefficients
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
87941
Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $
21726
Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
108318
counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
57418
How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
79652
Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$
116356
Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$
17491
Proof by Induction: Solving $1+3+5+\cdots+(2n-1)$
50599
Prove that $\frac{1}{1*3}+\frac{1}{3*5}+\frac{1}{5*7}+...+\frac{1}{(2n-1)(2n+1)}=\frac{n}{2n+1}$
49066
Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
82742
Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $
17714
Proof $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} = H_n$ by induction
34299
Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$