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

Step 1: Obtain the required equation.

Given

$S_5+S_7=167$

$S_{10}=235$

We know that the sum of term of an AP is given by the formula $S_n=\left(\frac{n}{2}\right)[2a+(n-1\left)d\right]$

Where,

$a$is first term

$a_n$ is the nth term

$d$ is the common difference

Substitute$S_{7}and{S}_5$ in $S_5+S_7=167$:

$$\begin{gathered} S_5+S_7=167...\left[\because S_n=\left(\frac{n}{2}\right)[2a+(n-1\left)d\right]\right] \\ \begin{array}{r}\Rightarrow \left(\frac{5}{2}\right)\left[2a+\left(5-1\right)d\right]+\left(\frac{7}{2}\right)\left[2a+\left(7-1\right)d\right]=167\\ \Rightarrow 5(2a+4d)+7(2a+6d)=334\\ \Rightarrow 10a+20d+14a+42d=334\\ \Rightarrow 24a+62d=334\\ \Rightarrow 12a+31d=167\\ \Rightarrow 12a=167-31d\end{array} \end{gathered}$$

Hence the equation obtained is $12\mathrm{a}-31\mathrm{d}=167...\left(i\right)$

Also given that,

$\begin{array}{r}{S}_{10}=235\\ \Rightarrow \left(\frac{10}{2}\right)[2a+(10-1\left)d\right]=235\\ \Rightarrow 5[2a+9d]=235\\ \Rightarrow 2a+9d=47\end{array}..\mathbf{.}\left[\because S_n=\left(\frac{n}{2}\right)[2a+(n-1\left)d\right]\right]$

Multiplying L.H.S and R.H.S by $6$ to obtain as follow:

$12a+54d=282...\left(ii\right)$

Step 2: Solve the obtained equation and determine the sum of the first twenty terms further.

Subtract $eq.\left(i\right)$ from $eq.\left(ii\right)$:

$\begin{array}{c}167-31d+54d=282\\ \Rightarrow 23d=115\\ \Rightarrow d=5\\ \end{array}$

And

$\begin{array}{c}12a=167-31\left(5\right)\\ \Rightarrow 12a=167-155\\ \Rightarrow 12a=12\\ \Rightarrow a=1\end{array}$

Substitute $a=1,d=5,n=20$ in $S_n=\left(\frac{n}{2}\right)[2a+(n-1\left)d\right]$ to find the sum of the first twenty terms:

:$$\begin{gathered} \begin{array}{r}{S}_{20}=\left(\frac{n}{2}\right)[2a+(20-1\left)d\right]\end{array} \\ \Rightarrow S_{20}=\frac{20}{2}\begin{array}{r}\left[2\right(1)+19(5\left)\right]\end{array} \\ \Rightarrow S_{20}=10[2+95] \\ \Rightarrow S_{20}=970 \end{gathered}$$

Therefore, the sum of first $20$ terms is $970$.