Question

The sum of the products of the $2n$ numbers $\pm 1,\pm 2,\pm 3.\dots ,\pm n$ taking two at a time is

• A. $\frac{n(n+1)}{2}$
• B. $-\frac{n(n+1)}{2}$
• C. $\frac{n(n+1)(2n+1)}{6}$
• D. $-\frac{n(n+1)(2n+1)}{6}$

Answer 1

Answer: (D)

We have,
$(1-1+2-2+3-3+\dots +n-n{)}^2$
$=1^2+1^2+2^2+2^2+\dots +n^2+n^2+25$
where $S$ is the required sum.
$\Rightarrow o=2\left(1^2+2^2+\dots +n^2\right)+2S$
$\Rightarrow S=-\left(1^2+2^2+\dots +n^2\right)$
$=-\frac{n(n+1)(2n+1)}{6}$