query: How to show $1+r+r^2 + \cdots + r^n = \frac{r^{n+1} - 1}{r-1}$ where $r \neq 1$
positive_passages.docid: 2678
positive_passages.text: Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
negative_passages.docid: 135747
negative_passages.text: Prove $x^{rs}-1=(x^s-1)(x^{s(r-1)}+...+x^s+1)$
negative_passages.docid: 63474
negative_passages.text: Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction?
negative_passages.docid: 93427
negative_passages.text: Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
negative_passages.docid: 129171
negative_passages.text: Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$
negative_passages.docid: 101057
negative_passages.text: Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
negative_passages.docid: 90043
negative_passages.text: Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + ... + r^{n-1} = 0$.
negative_passages.docid: 92810
negative_passages.text: What is $\sum_{r=1}^\infty\frac{r+2}{2^{r+1}(r)(r+1)}$?
negative_passages.docid: 146533
negative_passages.text: Having problems proving $ \sum_{r=0}^{n}(r+1) \binom{n}{r} = (n+2)2^{n-1} $
negative_passages.docid: 5927
negative_passages.text: How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
negative_passages.docid: 16812
negative_passages.text: Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
negative_passages.docid: 66082
negative_passages.text: Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
negative_passages.docid: 113593
negative_passages.text: How can I prove that $\frac{n^2}{x_1+x_2+\dots+x_n} \le \frac{1}{x_1}+ \frac{1}{x_2} +\dots+ \frac{1}{x_n}$?
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: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
negative_passages.docid: 43509
negative_passages.text: Evaluating $\sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)}$ for $x\in\mathbb{R}^{+}$
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} $