Let $x,y \in \mathbb{R}$..Show that $x^{n+1}-y^{n+1}=(x-y) \sum\limits_{k=0}^n x^k y^{n-k}$ for all $n \in \mathbb{N}_0$
69131
Different proofs of $\,a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?
123955
How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$?
79124
Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
84010
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
37569
Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
49771
Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.
2885
Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
155757
Number theory: $x^y + 1 = y^x$
87665
Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$
59475
Show that $\sqrt{n+1}-\sqrt{n}\to0$
20626
Prove that $x^y < y^x$
33082
Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
105971
Convergence of $\sum \limits_{n=1}^{\infty}\sqrt{n^3+1}-\sqrt{n^3-1}$