query: Prove by Induction (Geometric Progression): $\sum_{i=0}^n q^i=\frac{q^{n+1}-1}{q-1}$
positive_passages.docid: 26790
positive_passages.text: Proving the geometric sum formula by induction
negative_passages.docid: 146200
negative_passages.text: proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
negative_passages.docid: 100080
negative_passages.text: Proving with induction $(1-x)^n<\frac 1 {1+nx}$
negative_passages.docid: 57491
negative_passages.text: Double induction example: $ 1 + q + q^2 + q^3 + \cdots + q^{n-1} + q^n = \frac {q^{n+1}-1}{q-1} $
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: 17714
negative_passages.text: Proof $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k} = H_n$ by induction
negative_passages.docid: 123757
negative_passages.text: Proving a summation inequality with induction
negative_passages.docid: 112907
negative_passages.text: Math Induction Proof: $(1+\frac1n)^n < n$
negative_passages.docid: 121544
negative_passages.text: Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
negative_passages.docid: 92046
negative_passages.text: Proving $\sum\limits_{i=0}^n  i 2^{i-1} = (n+1) 2^n - 1$ by induction
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: 81136
negative_passages.text: Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$
negative_passages.docid: 36639
negative_passages.text: Prove by induction that $\sum_{r=0}^{n}\begin{pmatrix}n \\ r \end{pmatrix}=2^k$
negative_passages.docid: 40348
negative_passages.text: Proving by mathematical induction: $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}>2(\sqrt{n+1}-1)$
negative_passages.docid: 88292
negative_passages.text: Prove by induction that $\sum\limits_{i=1}^n \frac{1}{n+i} \leq \frac{3}{4}$
negative_passages.docid: 131771
negative_passages.text: Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$