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CIMA 2014 Problem 1

Source:

August 23, 2014
limitinductionlogarithmscollege contests

Problem Statement

Let {an}n≥1\{a_n\}_{n\geq 1} be a sequence of real numbers which satisfies the following relation: an+1=10nan2a_{n+1}=10^n a_n^2 (a) Prove that if a1a_1 is small enough, then lim⁡n→∞an=0\displaystyle\lim_{n\to\infty} a_n =0. (b) Find all possible values of a1∈Ra_1\in \mathbb{R}, a1≥0a_1\geq 0, such that lim⁡n→∞an=0\displaystyle\lim_{n\to\infty} a_n =0.
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