## Value of Sum

What is the value of the following sum?
$2006[(-1)^1+(-1)^2+(-1)^3+(-1)^4+\cdots+(-1)^{2006}]$
Source: NCTM Mathematics Teacher, September 2006

Solution
The square bracket in the sum expression contains 1003 pairs of consecutive powers $(-1)^n+(-1)^{n+1}$ which pairwise sum to zero
$(-1)^1+(-1)^2=-1+1=0$
$(-1)^3+(1-)^4=-1+1=0$
$(-1)^5+(-1)^6=-1+1=0$
$\cdots$
$(-1)^{2005}+(-1)^{2006}=-1+1=0$
$2006[(-1)^1+(-1)^2+(-1)^3+\cdots+(-1)^{2006}]=2006[0+0+0+\cdots+0]=0$

Answer: $0$