Question

The value of $1^{2}+3^{2}+5^{2}+............+25^{2} is:$

• A. 2925
• B. 1469
• C. 1728
• D. 1456

Answer 1

Answer: (A)

Consider $1^{2}+3^{2}+5^{2}+.........+25^{2} $
$n^{th} \, term\, T_{n}=\left(2n-1\right)^{2}, n=1, ....... 13 $
$Now, \, S_{n} =\sum\limits_{n=1}^{13} T_{n} =\sum\limits_{n=1}^{13} \left(2n-1\right)^{2}$
$ =\sum\limits_{n=1}^{13} 4n^{2} +\sum\limits_{n=1}^{13} 1-\sum\limits_{n=1}^{13} 4n $
$=4\sum n^{2}+13-4 \sum n$
$ =4\left[\frac{n \left(n+1\right)\left(2n+1\right)}{6}\right]+13-4 \frac{n\left(n+1\right)}{2}$
Put $n = 13$, we get
$S_{n}=26\times14\times9+13-26\times14$
= $3276+13-364$
= $2925$.