AMC12 2020 A

AMC12 2020 A · Q22

AMC12 2020 A · Q22. It mainly tests Functions basics, Complex numbers (rare).

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 $n \geq 0$, where $i = \sqrt{-1}$. What is $\sum_{n=0}^\infty \frac{a_n b_n}{7^n}$?

设$(a_n)$和$(b_n)$是实数序列，使得$(2 + i)^n = a_n + b_n i$对所有整数$n \geq 0$成立，其中$i = \sqrt{-1}$。求$\sum_{n=0}^\infty \frac{a_n b_n}{7^n}$？

(A) $\frac{3}{8}$ $\frac{3}{8}$

(B) $\frac{7}{16}$ $\frac{7}{16}$

(D) $\frac{9}{16}$ $\frac{9}{16}$

(E) $\frac{4}{7}$ $\frac{4}{7}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): It follows from the original equality that $$\left(\frac{2+i}{\sqrt{7}}\right)^{2n}=\frac{(a_n+b_ni)^2}{7^n}=\left(\frac{a_n^2-b_n^2}{7^n}\right)+\left(\frac{2a_nb_n}{7^n}\right)i.$$ Therefore $$\sum_{n=0}^{\infty}\frac{a_nb_n}{7^n}=\frac12\,\mathrm{Im}\left(\sum_{n=0}^{\infty}\left(\frac{2+i}{\sqrt{7}}\right)^{2n}\right)$$ $$=\frac12\,\mathrm{Im}\left(\sum_{n=0}^{\infty}\left(\frac{3+4i}{7}\right)^n\right)$$ $$=\frac12\,\mathrm{Im}\left(\frac{1}{1-\frac{3+4i}{7}}\right).$$ The preceding geometric series converges as shown because the modulus of the common ratio is less than 1. From $$\frac{1}{1-\frac{3+4i}{7}}=\frac{7}{4-4i}=\frac14\left(\frac{7}{1-i}\cdot\frac{1+i}{1+i}\right)=\frac78(1+i),$$ it follows that $$\sum_{n=0}^{\infty}\frac{a_nb_n}{7^n}=\frac{7}{16}.$$

答案（B）：由原等式可得 $$\left(\frac{2+i}{\sqrt{7}}\right)^{2n}=\frac{(a_n+b_ni)^2}{7^n}=\left(\frac{a_n^2-b_n^2}{7^n}\right)+\left(\frac{2a_nb_n}{7^n}\right)i.$$ 因此 $$\sum_{n=0}^{\infty}\frac{a_nb_n}{7^n}=\frac12\,\mathrm{Im}\left(\sum_{n=0}^{\infty}\left(\frac{2+i}{\sqrt{7}}\right)^{2n}\right)$$ $$=\frac12\,\mathrm{Im}\left(\sum_{n=0}^{\infty}\left(\frac{3+4i}{7}\right)^n\right)$$ $$=\frac12\,\mathrm{Im}\left(\frac{1}{1-\frac{3+4i}{7}}\right).$$ 上述等比级数收敛，因为公比的模小于 1。并且 $$\frac{1}{1-\frac{3+4i}{7}}=\frac{7}{4-4i}=\frac14\left(\frac{7}{1-i}\cdot\frac{1+i}{1+i}\right)=\frac78(1+i),$$ 从而 $$\sum_{n=0}^{\infty}\frac{a_nb_n}{7^n}=\frac{7}{16}.$$

Topics