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Sequence of reals x_1=3 and x_n={n+2}/{3n}(x_{n-1}+2).

Source: Vietnamese National Mathematical Olympiad 2012-P1

February 8, 2012
limitalgebra proposedalgebra

Problem Statement

Define a sequence {xn}\{x_n\} as: {x1=3xn=n+23n(xn−1+2)  for n≥2.\left\{\begin{aligned}& x_1=3 \\ & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right. Prove that this sequence has a finite limit as n→+∞.n\to+\infty. Also determine the limit.
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