query: Prove $2^{(n+1)}>n^2$ by induction
positive_passages.docid: 20964
positive_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 40692
negative_passages.text: Proving by strong induction that $\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 33757
negative_passages.text: For what natural numbers is $n^3 < 2^n$? Prove by induction
negative_passages.docid: 96332
negative_passages.text: Prove by induction that $\sqrt[n+1]{n+1}<\sqrt[n]{n}$ for $n\ge 3$
negative_passages.docid: 152577
negative_passages.text: Proving $4^n > n^4$ holds for $n\geq 5$ via induction.
negative_passages.docid: 89175
negative_passages.text: Proof of $(a+b)^{n+1}$
negative_passages.docid: 88055
negative_passages.text: Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$
negative_passages.docid: 52115
negative_passages.text: Show that $e^x > 1 + x + x^2/2! + \cdots + x^k/k!$ for $n \geq 0$, $x > 0$ by induction
negative_passages.docid: 44161
negative_passages.text: Using induction to prove $\sum_{k=0}^{N-1} 2^k = 2^N-1$
negative_passages.docid: 136274
negative_passages.text: $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
negative_passages.docid: 65391
negative_passages.text: Prove by induction. $n \in \mathbb N$.
negative_passages.docid: 54369
negative_passages.text: Proof of $n^2 \leq 2^n$.
negative_passages.docid: 153123
negative_passages.text: Proving by induction $4^n  0$
negative_passages.docid: 50670
negative_passages.text: Use Mathematical Induction to prove 6 divides $n(n+1)(2n+1)$
negative_passages.docid: 2554
negative_passages.text: Show that if $n>2$, then $(n!)^2>n^n$.