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Calculate the sum in the sequence

Source: Austrian Mathematical Olympiad 1998, Part 2, D1, P3

June 29, 2011

functioncombinatorics proposedcombinatorics

Problem Statement

Let

a_n

be a sequence recursively defined by

a_0 = 0, a_1 = 1

and

a_{n+2} = a_{n+1} + a_n

. Calculate the sum of

a_n\left( \frac 25\right)^n

for all positive integers

n

. For what value of the base

b

we get the sum

1