query: Prove by mathematical induction, or otherwise, that for all integers $n\ge 1$
positive_passages.docid: 9479
positive_passages.text: $\sum \cos$ when angles are in arithmetic progression
negative_passages.docid: 112907
negative_passages.text: Math Induction Proof: $(1+\frac1n)^n < n$
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
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: 66996
negative_passages.text: Proof by induction of Bernoulli's inequality $ (1+x)^n \ge 1+nx$
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: 49299
negative_passages.text: Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$.
negative_passages.docid: 86479
negative_passages.text: How to Prove with Mathematical Induction $3^n > n^2$
negative_passages.docid: 143968
negative_passages.text: Proving this inequality by mathematical induction
negative_passages.docid: 139269
negative_passages.text: Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
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: 8609
negative_passages.text: Prove by induction that $n!>2^n$
negative_passages.docid: 108596
negative_passages.text: Proof by induction for "sum-of"
negative_passages.docid: 25496
negative_passages.text: Avoiding proof by induction