Computing The Arithmetic-Harmonic Mean

Calculus Level 3

Define two sequences \(\{a_n\}\) and \(\{b_n\}\) by the recursive system

\[\begin{array} &a_0 = 4, &b_0 = 25, &a_{n+1} = \frac{a_n + b_n}{2}, &b_{n+1} = \frac{2}{\frac{1}{a_n} + \frac{1}{b_n}}.\end{array}\]

What is \(\displaystyle \lim_{n\to\infty} a_n?\)

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