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sequence with reciprocal of sum of squares

Source: Simon Marais 2017 A2

April 29, 2021

algebraSequences

Problem Statement

Let

a_1,a_2,a_3,\ldots

be the sequence of real numbers deﬁned by

a_1=1

and

a_m=\frac1{a_1^2+a_2^2+\ldots+a_{m-1}^2}\qquad\text{for }m\ge2.

Determine whether there exists a positive integer

N

such that

a_1+a_2+\ldots+a_N>2017^{2017}.

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