query: Proof of $\frac{x^n - a^n}{x-a} = x^{n-1} + x^{n-2}a + x^{n-3}a^3+...+xa^{n-2} + a^{n-1}$
positive_passages.docid: 144953
positive_passages.text: Value of $\frac{x^n -a^n}{ x-a}$?
negative_passages.docid: 30385
negative_passages.text: Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$
negative_passages.docid: 71547
negative_passages.text: Prove $ 1+\left( 1-\frac{x}{n}\right)+...+\left(1-\frac{x}{n}\right)^{n-1}=\frac{n}{x}\left(1-\left(1-\frac{x}{n}\right)^n\right)$
negative_passages.docid: 149700
negative_passages.text: Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
negative_passages.docid: 142132
negative_passages.text: How do you prove $\sum \frac {n}{2^n} = 2$?
negative_passages.docid: 34904
negative_passages.text: Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$
negative_passages.docid: 17100
negative_passages.text: Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
negative_passages.docid: 114418
negative_passages.text: A deduction from $x^n-1=(x^2-1)\prod_{k=1}^{(n-2)/2}[x^2-2x\cos{\frac{2k\pi}{n}}+1]$
negative_passages.docid: 145882
negative_passages.text: How prove this $\frac{a}{(bc-a^2)^2}+\frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2}=0$
negative_passages.docid: 120954
negative_passages.text: Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
negative_passages.docid: 15838
negative_passages.text: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
negative_passages.docid: 36906
negative_passages.text: Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
negative_passages.docid: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 148744
negative_passages.text: Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$
negative_passages.docid: 56101
negative_passages.text: Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
negative_passages.docid: 89265
negative_passages.text: Proving that $x - \frac{x^3}{3!} 0$
negative_passages.docid: 143236
negative_passages.text: $ S_{n}=\frac{x}{x+1}+\frac{x^2}{(x+1)(x^2+1)}+...........+\frac{x^{2^{n}}}{(x+1)(x^2+1)...(x^{2^{n}}+1)}$