query: Simple proof by induction: $1^3 + 2^3 +3^3 +...+ n^3 = [ (n(n+1))/2 ]^2 $
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: 121107
negative_passages.text: Proof by induction: $n^{n+1} > (n + 1)^n$, $(1 + x)^n \ge 1 + nx$, other inequalities
negative_passages.docid: 88055
negative_passages.text: Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$
negative_passages.docid: 70625
negative_passages.text: Prove By Induction $1 + 1/2 + 1/3 + ... + 1/(2^n) ≥ 1 + n/2$
negative_passages.docid: 146200
negative_passages.text: proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
negative_passages.docid: 6488
negative_passages.text: Proof that $n^3 + 3n^2 + 2n$ is a multiple of $3$.
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: 15838
negative_passages.text: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
negative_passages.docid: 70386
negative_passages.text: Proof by induction: $2^n > n^2$ for all integer $n$ greater than $4$
negative_passages.docid: 107824
negative_passages.text: Proof by induction that $a_0 = 1, a_1 = 1, a_n=2a_{n-1} + 3a_{n-2}$ satisfies $a_n = \frac12 (3^n) + \frac12 (-1)^n$
negative_passages.docid: 108596
negative_passages.text: Proof by induction for "sum-of"
negative_passages.docid: 101899
negative_passages.text: Proof By Induction of a Sum
negative_passages.docid: 6165
negative_passages.text: Proof by Induction $n^2+n$ is even
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$
negative_passages.docid: 138846
negative_passages.text: Prove by induction that $3\mid n^4-n^2$ for all $n \in \mathbb{Z^+}$, $n\ge2$.
negative_passages.docid: 62613
negative_passages.text: Need to proof by induction (calculus1 n and n+1)
negative_passages.docid: 115912
negative_passages.text: Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$