# Limit the mixed recurrence

Calculus Level 3

Let $$a_0 = 1, b_0 = 2,$$ and for $$n\geq 0$$ $a_{n+1} = \frac{a_n + b_n}{2},\quad b_{n+1} = \sqrt{a_{n+1}b_n}.$ If $$\displaystyle \lim_{n\to\infty } a_n b_n = \dfrac c{\pi^ d}$$ for positive integers $$c$$ and $$d$$, find $$c+d$$.

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