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Arithmetic progression

Source: Finnish Mathematics Competition 1999, Final Round, Problem 2

November 14, 2011
arithmetic sequencealgebra unsolvedalgebra

Problem Statement

Suppose that the positive numbers a1,a2,..,ana_1, a_2,.. , a_n form an arithmetic progression; hence ak+1−ak=d,a_{k+1}- a_k = d, for k=1,2,...,n−1.k = 1, 2,... , n - 1. Prove that 1a1a2+1a2a3+...+1an−1an=n−1a1an.\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.
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