Question

Sum $2,3\frac{1}{4},4\frac{1}{2}$, ..... to $20$ terms.

Answer 1

Given series is $2,3\frac{1}{4},4\frac{1}{2},.......$ upto 20 terms

The given series can be written as $\frac{8}{4},\frac{13}{4},\frac{18}{4},.....$ to 20 terms

$\Rightarrow \frac{1}{4}(8+13+18+......)$ to 20 terms

Observe that $8,13,\mathrm{18.....}$ is in Arithmetic Progression with $8$ as first term and $5$ as common difference.

So, the sum of first $20$ terms in that series will be $\frac{1}{4}\left(\frac{20}{2}\right(2\times 8+(20-1)\times 5\left)\right)=\frac{1}{4}\left(1110\right)=\frac{555}{2}$