Question

The sum $1^2-2 \cdot 3^2+3 \cdot 5^2-4 \cdot 7^2+5 \cdot 9^2-\ldots+15 \cdot 29^2$ is_____

Answer 1

Separating odd placed and even placed terms we get
$ S =\left(1.1^2+3.5^2+\ldots .15 .(29)^2\right)-\left(2.3^2+4.7^2\right. +\ldots .+14 .(27)^2$
$ S =\displaystyle\sum_{ n =1}^8(2 n -1)(4 n -3)^2-\displaystyle\sum_{ n =1}^7(2 n )(4 n -1)^2$
Applying summation formula we get
$=29856-22904=6952$