Can I prove this combinatorial relation using induction on $k$? $\sum\limits_{i = 0}^{k} \binom{n + i}{i} = \binom{n + k + 1}{k}$
8476
Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
67886
How to prove $\sum_{k=1}^{n}F_k = F_{n+2}-1$ by induction when $F_n$ is the Fibonacci sequence
92046
Proving $\sum\limits_{i=0}^n  i 2^{i-1} = (n+1) 2^n - 1$ by induction
10777
Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
143113
Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
67927
Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$
108596
Proof by induction for "sum-of"
33082
Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.
92440
Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
136580
Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$
89552
proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
142994
How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?
101899
Proof By Induction of a Sum
118571
Why is $\sum\limits_{k=0}^{n}(-1)^k\binom{n}{k}^2=(-1)^{n/2}\binom{n}{n/2}$ if $n$ is even?
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $