query: Prove by induction that $1^3 + 2^3 + 3^3 + .....+ n^3= \frac{n^2(n+1)^2}{4}$ for all $n\geq1$.
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: 130093
negative_passages.text: Prove by Induction : $n < 2^n$
negative_passages.docid: 87941
negative_passages.text: Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $
negative_passages.docid: 57491
negative_passages.text: Double induction example: $ 1 + q + q^2 + q^3 + \cdots + q^{n-1} + q^n = \frac {q^{n+1}-1}{q-1} $
negative_passages.docid: 71451
negative_passages.text: A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$
negative_passages.docid: 49299
negative_passages.text: Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$.
negative_passages.docid: 125618
negative_passages.text: Prove $2^{3^n}+1 $ is a multiple of $3^{n+1}$ by induction
negative_passages.docid: 151646
negative_passages.text: Prove by induction that $\sum_{i=1}^{n} i \cdot 2^i = (n-1) \cdot 2^{n+1} +2$. Help finding my mistake
negative_passages.docid: 95000
negative_passages.text: Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
negative_passages.docid: 21100
negative_passages.text: Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
negative_passages.docid: 38995
negative_passages.text: Induction proof that $\sum_{j=n}^{2n-1} (2j + 1) = 3n^2$ - what happened?
negative_passages.docid: 10629
negative_passages.text: How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction
negative_passages.docid: 88292
negative_passages.text: Prove by induction that $\sum\limits_{i=1}^n \frac{1}{n+i} \leq \frac{3}{4}$
negative_passages.docid: 3043
negative_passages.text: Prove $n!>2^n$ for $n\geq4$ using induction.
negative_passages.docid: 63652
negative_passages.text: Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$
negative_passages.docid: 148330
negative_passages.text: Induction Proof for $F_{2n} = F^2_{n+1} - F^2_{n-1}$
negative_passages.docid: 101899
negative_passages.text: Proof By Induction of a Sum