Q. The sum of the products of the numbers $\pm 1, \pm 2,\ldots \pm n$, taken two at a time is
A) $\frac{- n(n + 1)}{2}$
B) $\frac{n(n + 1)(2n + 1)}{6}$
C) $\frac{- n(n + 1)(2n + 1)}{6}$
D) 0
Explanation
(c)
Since $( \pm 1 \pm 2 \pm ... \pm n)^{2}$
$= 2\lbrack 1^{2} + 2^{2} + ... + n^{2}\rbrack$
$+ 2$[sum of product of two terms]
$\Rightarrow 2$[sum of product of two terms]
$= 0 - 2\left\lbrack \frac{n(n + 1)(2n + 1)}{6} \right\rbrack$
$\Rightarrow$ Sum of product of two terms
$= \frac{- n(n + 1)(2n + 1)}{6}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.