Arithmetic About Recursion

Probability Level 2

Let k1,k2,, k_1, k_2, \ldots, be a sequence that is recursively defined as kn+2=kn+1+2knk_{n+2} = k_{n+1} + 2 k_{n}, for all n1n\geq 1 , with k1=k2=1k_1 = k_2 = 1. The infinite sum, S=k171+k272+k373S = \frac{k_1}{7 ^1} + \frac{k_2}{7 ^2} + \frac{k_3}{7 ^3} \ldots, is a fraction of the form ab\frac{a}{b}, where aa and bb are coprime integers. What is the value of a+ba+b?

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