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Sum of Complex Recursive Series

Source: 2020 AMC 12A #22

January 31, 2020
AMCAMC 12AMC 12 A2020 AMC 12A2020 AMCseries

Problem Statement

Let (an)(a_n) and (bn)(b_n) be the sequences of real numbers such that (2+i)n=an+bni (2 + i)^n = a_n + b_ni for all integers n≥0n\geq 0, where i=−1i = \sqrt{-1}. What is ∑n=0∞anbn7n ?\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?
<spanclass=′latex−bold′>(A)</span>38<spanclass=′latex−bold′>(B)</span>716<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>916<spanclass=′latex−bold′>(E)</span>47<span class='latex-bold'>(A) </span>\frac 38\qquad<span class='latex-bold'>(B) </span>\frac7{16}\qquad<span class='latex-bold'>(C) </span>\frac12\qquad<span class='latex-bold'>(D) </span>\frac9{16}\qquad<span class='latex-bold'>(E) </span>\frac47
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