Question

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

Answer 1

Answer: 6952

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