query: Show that $(\sqrt{2}+1)^n = \sqrt{m}+\sqrt{m-1}$ for some positive integer $m$
positive_passages.docid: 114506
positive_passages.text: Expansion of $(1+\sqrt{2})^n$
negative_passages.docid: 15205
negative_passages.text: Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$
negative_passages.docid: 73277
negative_passages.text: Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
negative_passages.docid: 129433
negative_passages.text: $\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational
negative_passages.docid: 103854
negative_passages.text: Showing that $Q_n=D_n+D_{n-1}$
negative_passages.docid: 75373
negative_passages.text: Show that $2^n>n$ for $n\in \mathbb{N}$
negative_passages.docid: 42292
negative_passages.text: Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.
negative_passages.docid: 26876
negative_passages.text: How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?
negative_passages.docid: 90962
negative_passages.text: For positive integers $m,n$ if $\sqrt 7 - \frac{m}{n} > 0$ then prove that $\sqrt 7 - \frac{m}{n} > \frac{1}{{mn}}$
negative_passages.docid: 52427
negative_passages.text: Prove, $\sqrt{n} \le (n!)^\frac{1}{n} \le \frac{n+1}{2} $
negative_passages.docid: 110826
negative_passages.text: prove that : $ i = \sqrt {-1}\ $
negative_passages.docid: 25004
negative_passages.text: Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
negative_passages.docid: 150088
negative_passages.text: Prove that $\sqrt {n − 1} +\sqrt {n + 1}$ is irrational for every integer $n ≥ 1$
negative_passages.docid: 106767
negative_passages.text: $|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
negative_passages.docid: 25583
negative_passages.text: Show that the sequence $a_0 = 1$, $a_{n+1 }= \sqrt{2+a_n}$ is monotonically increasing
negative_passages.docid: 46147
negative_passages.text: Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
negative_passages.docid: 45037
negative_passages.text: Show $(2^m-1,2^n+1)=1$ if $m$ is odd