If the sum of the first 40 terms of the series,
Question:

If the sum of the first 40 terms of the series, $3+4+8+9+$ $13+14+18+19+\ldots$ is $(102) \mathrm{m}$, then $\mathrm{m}$ is equal to:

1. (1) 20

2. (2) 25

3. (3) 5

4. (4) 10

Correct Option: 1

Solution:

$S=\underbrace{3+4}+\underbrace{8+9}+\underbrace{13+14}+\underbrace{18+19} \ldots . .40$ terms

$S=7+17+27+37+47+\ldots .20$ terms

$S_{40}=\frac{20}{2}[2 \times 7+(19) 10]=10[14+190]$

$=10[2040]=(102)(20)$

$\Rightarrow \quad m=20$