Arithmetic About Recursion

Discrete Mathematics Level 2

Let \( k_1, k_2, \ldots, \) be a sequence that is recursively defined as \(k_{n+2} = k_{n+1} + 2 k_{n}\), for all \(n\geq 1 \), with \(k_1 = k_2 = 1\). The infinite sum, \(S = \frac{k_1}{7 ^1} + \frac{k_2}{7 ^2} + \frac{k_3}{7 ^3} \ldots\), is a fraction of the form \(\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. What is the value of \(a+b\)?