query: Prove the formula for the sum of consecutive cubes
positive_passages.docid: 16812
positive_passages.text: Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
negative_passages.docid: 21000
negative_passages.text: Calculating $\sum_{k=1}^nk(k!)$ combinatorially
negative_passages.docid: 127855
negative_passages.text: Show that the product of any $m$ consecutive positive integers is divisible by $m!$
negative_passages.docid: 139183
negative_passages.text: How can I prove the formula for calculating successive entries in a given row of Pascal's triangle?
negative_passages.docid: 94389
negative_passages.text: Proving formula for product of first n odd numbers
negative_passages.docid: 135914
negative_passages.text: Evaluate the sum of cubes
negative_passages.docid: 8817
negative_passages.text: Closed form recurrence formula for getting N consecutive heads on a coin
negative_passages.docid: 11983
negative_passages.text: Proof of Clarkson's Inequality
negative_passages.docid: 90904
negative_passages.text: Combinatorial proof of an identity with squared binomial coefficients
negative_passages.docid: 92106
negative_passages.text: Prove that $n$ is a sum of two squares?
negative_passages.docid: 15452
negative_passages.text: Proving determinant product rule combinatorially
negative_passages.docid: 107433
negative_passages.text: Prove the sum to n terms of the series
negative_passages.docid: 56828
negative_passages.text: A combinatorial proof that the alternating sum of binomial coefficients is zero
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 58428
negative_passages.text: Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
negative_passages.docid: 345
negative_passages.text: Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
negative_passages.docid: 143540
negative_passages.text: How to prove this decomposition of sum of squares?