query: Prove by induction: $n! \ge 2^{(n-1)}$ for any $n \ge 1$
positive_passages.docid: 16798
positive_passages.text: Prove the inequality $n! \geq 2^n$ by induction
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 18301
negative_passages.text: Proof by Induction involving Inequality and Factorials as denominators
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$
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: 101899
negative_passages.text: Proof By Induction of a Sum
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
negative_passages.docid: 108596
negative_passages.text: Proof by induction for "sum-of"
negative_passages.docid: 88055
negative_passages.text: Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$
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: 145519
negative_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
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: 10976
negative_passages.text: Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$
negative_passages.docid: 62556
negative_passages.text: Prove $3|n(n+1)(n+2)$ by induction
negative_passages.docid: 131771
negative_passages.text: Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
negative_passages.docid: 66358
negative_passages.text: Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$?
negative_passages.docid: 49299
negative_passages.text: Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$.