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a_n >= 1/n if a_{n+1}^2 + a_{n+1} = a_n, a_1=1 , a_i>=0

Source: Canadian Junior Mathematical Olympiad - CJMO 2020 p1

July 15, 2020

inequalitiesalgebraSequencerecurrence relation

Problem Statement

Let

a_1, a_2, a_3, . . .

be a sequence of positive real numbers that satisfies

a_1 = 1

and

a^2_{n+1} + a_{n+1} = a_n

for every natural number

n

. Prove that

a_n \ge \frac{1}{n}

for every natural number

n