query: Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
positive_passages.docid: 58633
positive_passages.text: Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
negative_passages.docid: 79652
negative_passages.text: Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$
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: 67886
negative_passages.text: How to prove $\sum_{k=1}^{n}F_k = F_{n+2}-1$ by induction when $F_n$ is the Fibonacci sequence
negative_passages.docid: 79657
negative_passages.text: Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$
negative_passages.docid: 1258
negative_passages.text: Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$
negative_passages.docid: 155897
negative_passages.text: Proof by induction - is this correct?
negative_passages.docid: 62747
negative_passages.text: Use induction to prove a product of sums of squares is a sum of squares
negative_passages.docid: 123427
negative_passages.text: Proving the sum of the first $n$ natural numbers by induction
negative_passages.docid: 138683
negative_passages.text: Proving $\prod_{i=2}^{i=n} \left(1-\frac{1}{i^2}\right) = \frac{n+1}{2n}$ by induction
negative_passages.docid: 151646
negative_passages.text: Prove by induction that $\sum_{i=1}^{n} i \cdot 2^i = (n-1) \cdot 2^{n+1} +2$. Help finding my mistake
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: 20271
negative_passages.text: Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
negative_passages.docid: 122019
negative_passages.text: Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
negative_passages.docid: 125618
negative_passages.text: Prove $2^{3^n}+1 $ is a multiple of $3^{n+1}$ by induction
negative_passages.docid: 123757
negative_passages.text: Proving a summation inequality with induction
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