2020 AMC 12A Problem 22

Problem 22 of 25HarderAlgebra

Let

(a_n)

and

(b_n)

be the sequences of real numbers such that

(2 + i)^n = a_n + b_n i

for all integers

n \ge 0,

where

i = \sqrt{-1}.

What is

\sum_{n=0}^{\infty} \frac{a_n b_n}{7^n}?

Show solution

Solution

Since

(a_n + b_n i)^2 = a_n^2 - b_n^2 + 2 a_n b_n i,

we have

a_n b_n

= \tfrac12 \operatorname{Im}\big((2 + i)^{2n}\big)

= \tfrac12 \operatorname{Im}\big((3 + 4i)^n\big).

Therefore the sum is

\tfrac12 \operatorname{Im} \displaystyle\sum_{n=0}^{\infty} \left(\frac{3 + 4i}{7}\right)^n

= \tfrac12 \operatorname{Im}\!\left(\frac{1}{1 - \frac{3 + 4i}{7}}\right).

This equals

\tfrac12 \operatorname{Im}\!\left(\dfrac{7}{4 - 4i}\right)

= \tfrac12 \operatorname{Im}\!\left(\dfrac{7(4 + 4i)}{32}\right)

= \tfrac12 \cdot \dfrac{28}{32}

= \dfrac{7}{16}.

Thus, B is the correct answer.