Computing The Arithmetic-Harmonic Mean

Calculus Level 3

Define two sequences {an}\{a_n\} and {bn}\{b_n\} by the recursive system

a0=4,b0=25,an+1=an+bn2,bn+1=21an+1bn.\begin{array}{c}&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 limnan?\displaystyle \lim_{n\to\infty} a_n?

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