Computing The Arithmetic-Harmonic Mean

Calculus Level 3

Define two sequences \(\{a_n\}\) and \(\{b_n\}\) by the recursive system: \(a_0 = 4\), \(b_0 = 25\), \[a_{n+1} = \frac{a_n + b_n}{2} \] \[b_{n+1} = \frac{2}{\frac{1}{a_n} + \frac{1}{b_n}}\] What is \(\displaystyle \lim_{n\to\infty} a_n\)?

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