query: Proving Identity In Combinatorial and Algebraic Way
positive_passages.docid: 16347
positive_passages.text: Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$
negative_passages.docid: 146053
negative_passages.text: Proving identities (mod $pq$) using Fermat's little theorem?
negative_passages.docid: 156564
negative_passages.text: Prove the identity $\sum_{{{\underset{k-even}{k=0}}}}^{n}{n \choose k}2^{k}=\frac{3^{n}+(-1)^{n}}{2}$
negative_passages.docid: 8970
negative_passages.text: Proving the cross product matrix tranformation identity with an alternative solution
negative_passages.docid: 113119
negative_passages.text: Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ
negative_passages.docid: 27585
negative_passages.text: Combinatorial proof of Stirling Number Identity (First Kind)
negative_passages.docid: 32791
negative_passages.text: Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$
negative_passages.docid: 116356
negative_passages.text: Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$
negative_passages.docid: 107021
negative_passages.text: Finding a combinatorial argument for an interesting identity: $\sum_k \binom nk \binom{m+k}n = \sum_i \binom ni \binom mi 2^i$
negative_passages.docid: 129304
negative_passages.text: Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$
negative_passages.docid: 109648
negative_passages.text: Proving an identity for Bernoulli polynomials
negative_passages.docid: 65983
negative_passages.text: Combinatorial proof of binomial coefficient summation identity
negative_passages.docid: 38206
negative_passages.text: Prove Pascal's Rule Algebraically
negative_passages.docid: 76635
negative_passages.text: Proof through combinatorial argument
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: 4610
negative_passages.text: Proving an identity with a combinatorial proof: $\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r}$
negative_passages.docid: 156383
negative_passages.text: Proving a combinatorial identity without double counting