Problem:
Let f(n)=(2−1+i3)n+(2−1−i3)n, where i=−1. What is f(2022) ?
Answer Choices:
A. −2
B. −1
C. 0
D. 3
E. 2
Solution:
Note that
2−1+i3=cos32π+isin32π=e32πi
and
2−1−i3=cos3−2π+isin3−2π=e3−2πi
By de Moivre's Formula,
f(n)=e32nπi+e3−2nπi
Therefore
f(2022)=e1348πi+e−1348πi
Because e2πi=1, the given expression equals 1+1=(E)2.
The problems on this page are the property of the MAA's American Mathematics Competitions