query: Prove $\sum_{k=0}^n (k + 1)\binom{n}{k}= 2^{n - 1}(n + 2)$
positive_passages.docid: 146533
positive_passages.text: Having problems proving $ \sum_{r=0}^{n}(r+1) \binom{n}{r} = (n+2)2^{n-1} $
negative_passages.docid: 13236
negative_passages.text: Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$
negative_passages.docid: 28027
negative_passages.text: Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
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: 99249
negative_passages.text: Sum of product of binomial coefficients: $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n + k}{k} = (-1)^n$
negative_passages.docid: 108318
negative_passages.text: counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
negative_passages.docid: 1139
negative_passages.text: Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
negative_passages.docid: 130223
negative_passages.text: Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
negative_passages.docid: 65672
negative_passages.text: Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?
negative_passages.docid: 79124
negative_passages.text: Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$
negative_passages.docid: 130543
negative_passages.text: If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
negative_passages.docid: 17714
negative_passages.text: Proof $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} = H_n$ by induction
negative_passages.docid: 92440
negative_passages.text: 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}$
negative_passages.docid: 116461
negative_passages.text: Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms
negative_passages.docid: 52113
negative_passages.text: Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.