Difference between revisions of "2020 AMC 12A Problems/Problem 22"
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Revision as of 17:51, 1 February 2020
Problem
Let 
and 
be the sequences of real numbers such that
![\[ (2 + i)^n = a_n + b_ni \]](http://latex.artofproblemsolving.com/9/9/1/9917b4f9dba198e73634715eaa0ff52b7e9e3ba5.png)
for all integers 
, where 
. What is![\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]](http://latex.artofproblemsolving.com/a/3/a/a3a51079920de0ca06d385c58399f194a5ff90cb.png)
Solution
Square the given equality to yield
![\[(3 + 4i)^n = (a_n + b_ni)^2 = (a_n^2 - b_n^2) + 2a_nb_ni,\]](http://latex.artofproblemsolving.com/e/a/f/eafed2044252dfe02ba40174e3fdf1321f502761.png)
so 
and
![\[\sum_{n\geq 0}\frac{a_nb_n}{7^n} = \frac12\operatorname{Im}\left(\sum_{n\geq 0}\frac{(3+4i)^n}{7^n}\right) = \frac12\operatorname{Im}\left(\frac{1}{1 - \frac{3 + 4i}7}\right) = \boxed{\frac 7{16}}.\]](http://latex.artofproblemsolving.com/7/f/8/7f85a857c032580b71599a24b712617c3a49855c.png)
See Also
| 2020 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
