# 2012-5 Iterative geometric mean

For given positive real numbers $$a_1,\ldots,a_k$$ and for each integer n≥k, let $$a_{n+1}$$ be the geometric mean of $$a_n, a_{n-1}, a_{n-2}, \ldots, a_{n-k+1}$$. Prove that $$\lim_{n\to\infty} a_n$$ exists and compute this limit.

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