query: Induction proof concerning a sum of binomial coefficients: $\sum_{j=m}^n\binom{j}{m}=\binom{n+1}{m+1}$
positive_passages.docid: 36786
positive_passages.text: Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
negative_passages.docid: 25054
negative_passages.text: Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction
negative_passages.docid: 89576
negative_passages.text: Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$
negative_passages.docid: 46132
negative_passages.text: Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?
negative_passages.docid: 131771
negative_passages.text: Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
negative_passages.docid: 10917
negative_passages.text: Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
negative_passages.docid: 60860
negative_passages.text: Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 15164
negative_passages.text: Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
negative_passages.docid: 136580
negative_passages.text: Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$
negative_passages.docid: 38995
negative_passages.text: Induction proof that $\sum_{j=n}^{2n-1} (2j + 1) = 3n^2$ - what happened?
negative_passages.docid: 75910
negative_passages.text: Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
negative_passages.docid: 79652
negative_passages.text: Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$
negative_passages.docid: 113321
negative_passages.text: Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
negative_passages.docid: 1167
negative_passages.text: How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
negative_passages.docid: 63510
negative_passages.text: Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
negative_passages.docid: 10777
negative_passages.text: Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$