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limit of squence

Source: Problem 2 VMO 2013 day 1

January 11, 2013
limitfunctionalgebra proposedalgebra

Problem Statement

Define a sequence {an}\{a_n\} as: {a1=1an+1=3−an+22an  for n≥1.\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.
Prove that this sequence has a finite limit as n→+∞n\to+\infty . Also determine the limit.
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