Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that $\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.
Source: Kosovo MO 2010 Grade 11, Problem 2
Tags: algebra
Let $a_1,a_2,...,a_n$ be an arithmetic progression of positive real numbers. Prove that $\tfrac {1}{\sqrt a_1+\sqrt a_2}+\tfrac {1}{\sqrt a_2+\sqrt a_3}+...+\tfrac {1}{\sqrt a_{n-1}+\sqrt a_n}=\tfrac{n-1}{\sqrt {a_1}+\sqrt{a_n}}$.