Question

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

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

Answer 1

Answer: (D)

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