query: Prove by mathematical induction and determine when there is an equality
positive_passages.docid: 16524
positive_passages.text: Proof of an inequality by induction: $(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + x_n$
negative_passages.docid: 120093
negative_passages.text: Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
negative_passages.docid: 112907
negative_passages.text: Math Induction Proof: $(1+\frac1n)^n < n$
negative_passages.docid: 121544
negative_passages.text: Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
negative_passages.docid: 71918
negative_passages.text: Mathematical Induction
negative_passages.docid: 30783
negative_passages.text: Prove inequality $n<3^n$ using mathematical induction
negative_passages.docid: 121874
negative_passages.text: Use Induction to prove that for all $n \in \mathbb{N}, (x^n + \frac{1}{x^n}) \in \mathbb{Z}$ if $x+\frac{1}{x}\in\mathbb{Z}$.
negative_passages.docid: 51730
negative_passages.text: How to prove this inequality $3^{n}\geq n^{2}$ for $n\geq 1$ with mathematical induction?
negative_passages.docid: 32427
negative_passages.text: Basic mathematical induction regarding inequalities
negative_passages.docid: 70625
negative_passages.text: Prove By Induction $1 + 1/2 + 1/3 + ... + 1/(2^n) ≥ 1 + n/2$
negative_passages.docid: 95000
negative_passages.text: Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
negative_passages.docid: 138036
negative_passages.text: Proving Inequality using Induction $a^n-b^n \leq na^{n-1}(a-b)$
negative_passages.docid: 124584
negative_passages.text: How to proof that $\forall n \ge 1: \sum_{i=1}^n (i(i+1))^{-1} = n(n+1)^{-1}$ using mathematical induction
negative_passages.docid: 134202
negative_passages.text: Proof by Induction - Math
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$
negative_passages.docid: 91197
negative_passages.text: Proving $n! \ge 2^{n-1 }$for all $n\ge1 $by mathematical Induction
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$