Question

If $S=1\left(25\right)+2\left(24\right)+3\left(23\right)+......+24\left(2\right)+25\left(1\right)$ , then the value of $\frac{S}{900}$ is equal to

Answer 1

$S=\sum_{r=1}^{25} r(26-r)=26 \sum_{r=1}^{25} r-\sum_{r=1}^{25} r^{2}$
$=26 \times \frac{25 \times 26}{2}-\frac{25 \times 26 \times 51}{6}$
$=\frac{25 \times 26}{2}\left(26-\frac{51}{3}\right)=\frac{25 \times 2613}{4} \times 9$
$\Rightarrow \frac{S}{900}=\frac{13}{4}=3.25$