query: proving that $(n-1)^n>n^{n-1}$
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: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 115799
negative_passages.text: Proving $x^{\log n} = n^{\log x}$
negative_passages.docid: 119118
negative_passages.text: Prove $n\mid \phi(2^n-1)$
negative_passages.docid: 76696
negative_passages.text: Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$
negative_passages.docid: 2197
negative_passages.text: Induction principle for $n!<n^n$
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
negative_passages.docid: 127293
negative_passages.text: Prove that $(a-b) \mid (a^n-b^n)$
negative_passages.docid: 26514
negative_passages.text: Prove that if $n$ is a composite, then $2^n-1$ is composite.
negative_passages.docid: 15838
negative_passages.text: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
negative_passages.docid: 20964
negative_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 145891
negative_passages.text: Prove that $n! > \sqrt{n^n}, n \geq 3$
negative_passages.docid: 10604
negative_passages.text: Prove $3^n > n^2$ by induction
negative_passages.docid: 87665
negative_passages.text: Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$
negative_passages.docid: 67291
negative_passages.text: Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
negative_passages.docid: 121107
negative_passages.text: Proof by induction: $n^{n+1} > (n + 1)^n$, $(1 + x)^n \ge 1 + nx$, other inequalities
negative_passages.docid: 44161
negative_passages.text: Using induction to prove $\sum_{k=0}^{N-1} 2^k = 2^N-1$