query: How to prove by induction that $(n+1)^n < n^{n+1}$ for $n\ge 3$?
positive_passages.docid: 26912
positive_passages.text: Prove by induction that for all $n \geq 3$: $n^{n+1} > (n+1)^n$
negative_passages.docid: 61117
negative_passages.text: Prove by induction: $n! \ge 2^{(n-1)}$ for any $n \ge 1$
negative_passages.docid: 108382
negative_passages.text: Prove $x^n-1$ is divisible by $x-1$ by induction
negative_passages.docid: 115459
negative_passages.text: Proving this $F_{n+1} \cdot F_{n-1} - F^2_n = (-1)^n$ by induction
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
negative_passages.docid: 82774
negative_passages.text: Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
negative_passages.docid: 70625
negative_passages.text: Prove By Induction $1 + 1/2 + 1/3 + ... + 1/(2^n) ≥ 1 + n/2$
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: 87379
negative_passages.text: How to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction?
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: 79145
negative_passages.text: Proving by induction: $2^n > n^3 $ for any natural number $n > 9$
negative_passages.docid: 102738
negative_passages.text: Show that $e^n>\frac{(n+1)^n}{n!}$ without using induction.
negative_passages.docid: 16798
negative_passages.text: Prove the inequality $n! \geq 2^n$ by induction
negative_passages.docid: 136274
negative_passages.text: $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
negative_passages.docid: 114867
negative_passages.text: Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
negative_passages.docid: 10629
negative_passages.text: How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$