2020 AMC 12A Problems/Problem 22

Problem

Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that $(2 + i)^n = a_n + b_ni$ for all integers $n\geq 0$ , where $i = \sqrt{-1}$ . What is $\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?$ $\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac9{16}\qquad\textbf{(E) }\frac47$

Solution

Square the given equality to yield $(3 + 4i)^n = (a_n + b_ni)^2 = (a_n^2 - b_n^2) + 2a_nb_ni,$ so $a_nb_n = \tfrac12\operatorname{Im}((3+4i)^n)$ and $\sum_{n\geq 0}\frac{a_nb_n}{7^n} = \frac12\operatorname{Im}\left(\sum_{n\geq 0}\frac{(3+4i)^n}{7^n}\right) = \frac12\operatorname{Im}\left(\frac{1}{1 - \frac{3 + 4i}7}\right) = \boxed{\frac 7{16}}.$

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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2020 AMC 12A Problems/Problem 22

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