query: Prove that for a natural number $n$ is true $1^3 + 2^3 +...+ n^3 = (1 + 2 +...+ n)^2$
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: 4333
negative_passages.text: Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$
negative_passages.docid: 62556
negative_passages.text: Prove $3|n(n+1)(n+2)$ by induction
negative_passages.docid: 54055
negative_passages.text: Proof $\prod_{i = 1}^n \frac{n + i}{2i-3} = 2^n(1-2n)$ using inducction
negative_passages.docid: 136868
negative_passages.text: Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$
negative_passages.docid: 31698
negative_passages.text: $f(m^2 + f(n)) = f(m)^2 + n$ for all natural numbers
negative_passages.docid: 44247
negative_passages.text: For $a^3+b^3+c^3=3$ prove that $a^4b+b^4c+c^4a\leq3$
negative_passages.docid: 142992
negative_passages.text: Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
negative_passages.docid: 4077
negative_passages.text: Using induction prove $n^3-n$ is divisible by 3 whenever n is a positive number.
negative_passages.docid: 81136
negative_passages.text: Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$
negative_passages.docid: 83862
negative_passages.text: How to prove that the number $1!+2!+3!+...+n!$ is never square?
negative_passages.docid: 122019
negative_passages.text: Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
negative_passages.docid: 154475
negative_passages.text: Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
negative_passages.docid: 152039
negative_passages.text: Prove that $a^{2^n}=1 \mod 2^{n+2}$
negative_passages.docid: 1258
negative_passages.text: Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$
negative_passages.docid: 153281
negative_passages.text: Prove by induction that $3^{2n+3}+40n-27$ is divisible by 64 for all n in natural numbers
negative_passages.docid: 123427
negative_passages.text: Proving the sum of the first $n$ natural numbers by induction