KöMaL 2011 C1060

Let $n$ denote a number divisible by 3. Every third term of the sequence $n$-1, $n$-2, ..., 2, 1 is cancelled. The subsequent pairs of terms of the remaining sequence are given alternating plus and minus signs: +($n$-1), +($n$-2), -($n$-4), -($n$-5), ... . Prove that the sum of the numbers obtained in this way is $n$.