Class 11

Math

Algebra

Sequences and series

If the sum of $n$ terms of an A.P. is $cn(n−1)$ where $c=0$, then sum of the squares of these terms is

- $c_{2}n_{2}(n+1)_{2}$
- $32 c_{2}n(n−1)(2n−1)$
- $32 c_{2}n(n+1)(2n+1)$
- None of these

Let the $A.P$ be $a,a+d,a+2d,…,a+(n−1)d$

Given $S_{n}=cn(n−1)$

$cn(n−1)=2n (2a+(h−1)d)$

$2c(n−1)=2a+(n−1)d$

$2c=d$

$−2c=2a−d$

$−2c=2a−2c$

$a=0$

Required series

$o,d_{2},(2d)_{2},........((n−1)d)_{2}$

Sum of $(n−1)$ terms square

$=o+d_{2}+(2d)_{2}+........+((n−1)d)_{2}$

$=d_{2}[1_{2}+2_{2}+3_{2}+⋯+(n−1)_{2}]$

$=∑[(n−1)×d]_{2}$

$=∑[(n−1)×2c]_{2}$

$=4c_{2}×∑ (n−1)_{2}$

$=64c_{2}×(n−1)n(2n−1) $ since sum of square of first n natural numbers $S=6n(n−1)(2n−1) $