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Limit of a sequence

Source: Moldova TST 2013

April 16, 2013
limitlogarithmsalgebra unsolvedalgebra

Problem Statement

Consider a positive real number aa and a positive integer mm. The sequence (xk)k∈Z+(x_k)_{k\in \mathbb{Z}^{+}} is defined as: x1=1x_1=1, x2=ax_2=a, xn+2=xn+1mxnm+1x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}. a)a) Prove that the sequence is converging. b)b) Find lim⁡n→∞xn\lim_{n\rightarrow \infty}{x_n}.
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