Question

The $4^{th}$ term of an AP is 9 and the $9^{th}$ term of the AP is 34. The sum of the first 10 terms of the AP is ___

Answer 1

Given, the $4^{th}$ term is 9 and the $9^{th}$ term is 34 in an AP
If the first term is taken as $a$, and the common difference is taken as $d$, then
$\Rightarrow a_4=a+(4-1)d=9$
$\Rightarrow a_9=a+(9-1)d=34$
From the above equations, we get d = 5 and a = - 6.
Now, sum of the AP upto 10 terms = $S_{10}$
$\Rightarrow S_{10}=\frac{10}{2}\left[2\right(-6)+(10-1\left)5\right]=5(-12+45)=165$