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a_m = s a_{m-1} + b_m, Sum a^2_i <= 100/(1-s^2)

Source: Norwegian Mathematical Olympiad 2012 - Abel Competition p4b

September 4, 2019

algebrainequalities

Problem Statement

Positive numbers

b_1, b_2,..., b_n

are given so that

b_1 + b_2 + ...+ b_n \le 10

. Further,

a_1 = b_1

and

a_m = sa_{m-1} + b_m

for

m > 1

, where

0 \le s < 1

. Show that

a^2_1 + a^2_2 + ... + a^2_n \le \frac{100}{1 - s^2}