Question

In an A.P., if the $5th$ and $12th$ terms are $30$ and $65$ respectively, what is the sum of first $20$ terms?

Answer 1

Let the $1^{st}$ term of $A.P$ is $=a$

Ans the common difference is $=d$

Given, $T_{5^{th}}=30=a+4d.....\left(1\right)$

$T_{{12}^{th}}=69=a+11d.....\left(2\right)$

Eqn $\left(2\right)-\left(1\right)$

$35=7d$

$d=5$

From Eqn $\left(1\right)$

$30=a+4\left(5\right)$

$a=10$

Now, the sum of $1^{st}20$ terms $=S_{20}$

$S_{20}=\frac{20}{2}[2a+(2a-1d]$

$\Rightarrow S_{20}=\frac{20}{2}\left[2\right(10)+19(5\left)\right]$

$\Rightarrow S_{20}=1150$