Question

The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms.

Answer 1

Let $S_n$ be the sum of n terms of an AP with first term a and common difference d. Then
$S_n=\frac{n}{2}(2a+(n-1\left)d\right)$
putting n=5 in $S_n$, we get
$S_5=\frac{5}{2}(2a+4d)\Rightarrow S_5=5(a+2d)$ ...(1)
putting n=7 in $S_n$, we get
$S_7=\frac{7}{2}(2a+6d)\Rightarrow S_7=7(a+3d)$ ...(2)
Adding (1) and (2), we get
$12a+31d=167$ ...(3)
Now,$S_{10}=235\Rightarrow \frac{10}{2}(2a+9d)=235$
$\Rightarrow 2a+9d=47$ ...(4)
On applying $6\times \left(4\right)-\left(3\right)$, we get
$23d=115\Rightarrow d=5$
Putting $d=5$ in (3), we get
$12a=167-31\times 5\Rightarrow a=1$
Therefore, Required sum $S_{20}=\frac{20}{2}[2\times 1+(20-1)\times 5$
$\Rightarrow S_{20}=10[2+95]$
$\Rightarrow S_{20}=970$