TITLE: Prove that $n$ is also a power of $2$.
QUESTION [8 upvotes]: An arithmetic progression consists of integers. The sum of the first $n$ terms of this progression is a power of two. 
Prove that $n$ is also a power of two.
Source :http://www.math.ucla.edu/~radko/circles/lib/data/Handout-967-1026.pdf

REPLY [3 votes]: $$S_n=\frac{n}{2}\left(a + a_n\right) = 2^k$$
$$\log_2 (S_n)=\log_2(n) + \log_2(a+a_n) = {k+1}$$
Since $k+1$ is an integer, $\log_2(n)$ must be an integer as well. Hence, $n$ is a power of $2$.