IMC 2015/3

Consider the recurrence relation with initial values $$F(0) = 0, F(1) = \frac{3}{2}$$ and $F(n) = \frac{5}{2} F(n-1) - F(n-2).$

To 2 decimal places, evaluate $\sum_{n=0}^ \infty \frac{1}{ F(2 ^n ) } .$

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