query: Prove $9 \mathrel| (4^n+6n-1)$ by induction
positive_passages.docid: 124240
positive_passages.text: Inductive proof of $\,9 \mathrel| 4^n+6n-1\,$ for all $\,n\in\Bbb N$
negative_passages.docid: 20271
negative_passages.text: Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
negative_passages.docid: 11642
negative_passages.text: Proving by induction that $2^n \le 2^{n+1}-2^{n-1} - 1$ . Does my proof make sense?
negative_passages.docid: 119436
negative_passages.text: Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
negative_passages.docid: 123427
negative_passages.text: Proving the sum of the first $n$ natural numbers by induction
negative_passages.docid: 134149
negative_passages.text: $z^n=(z+1)^n=1$, show that $n$ is divisible by $6$.
negative_passages.docid: 3043
negative_passages.text: Prove $n!>2^n$ for $n\geq4$ using induction.
negative_passages.docid: 66358
negative_passages.text: Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$?
negative_passages.docid: 27580
negative_passages.text: Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$
negative_passages.docid: 156356
negative_passages.text: Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction
negative_passages.docid: 6165
negative_passages.text: Proof by Induction $n^2+n$ is even
negative_passages.docid: 150462
negative_passages.text: Prove by induction that for every $n \ge 0$ the term $2^{2n+1} - 9n^2 + 3n -2$ is divisible by $54$
negative_passages.docid: 16177
negative_passages.text: Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
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: 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: 86479
negative_passages.text: How to Prove with Mathematical Induction $3^n > n^2$
negative_passages.docid: 44161
negative_passages.text: Using induction to prove $\sum_{k=0}^{N-1} 2^k = 2^N-1$