query: Is there a general form for $a_n=\sqrt{2+a_{n-1}}$?
positive_passages.docid: 14386
positive_passages.text: Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$
negative_passages.docid: 129454
negative_passages.text: Solution to a quadratic form
negative_passages.docid: 9920
negative_passages.text: $(n^2 - 1)!$ is a multiple of $(n!)^n$
negative_passages.docid: 9490
negative_passages.text: Let $(1+x+x^2)^n=a_0+a_1x+a_2x^2+\cdots+a_{2n}x^{2n}$ be an identity in $x$. Find $a_0 +a_2 +a_4+ \cdots +a_{2n} $ in terms of $n$
negative_passages.docid: 54900
negative_passages.text: How to prove $|\sum_{i=1}^n a_i|\le \sqrt{n} \sqrt{\sum_{i=1}^n a_i^2}$
negative_passages.docid: 2776
negative_passages.text: What can be said about the integral part of $(a+\sqrt{b})^n $
negative_passages.docid: 104917
negative_passages.text: Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?
negative_passages.docid: 78309
negative_passages.text: On an expansion of $(1+a+a^2+\cdots+a^n)^2$
negative_passages.docid: 143207
negative_passages.text: What is $\sum\limits_{i=1}^n \sqrt i\ $?
negative_passages.docid: 50930
negative_passages.text: Proving the continued fraction representation of $\sqrt{2}$
negative_passages.docid: 128608
negative_passages.text: Closed form for the series $a+a^p+(a^p)^p+\dots$
negative_passages.docid: 69136
negative_passages.text: factorization of a^n+1?
negative_passages.docid: 4310
negative_passages.text: Range of $(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$ is
negative_passages.docid: 39642
negative_passages.text: Efficient way to find $[\mathbb{Q}(\sqrt{5}, \sqrt{3}, \sqrt{2}): \mathbb{Q}(\sqrt{3}, \sqrt{2})]$
negative_passages.docid: 23726
negative_passages.text: A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$
negative_passages.docid: 73696
negative_passages.text: Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$
negative_passages.docid: 73277
negative_passages.text: Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$