Question

The sum of the first $7$ terms of an AP is $63$ and the sum of it's next $7$ terms is $161$. Find the ${28}^{th}$ term of this AP.

Answer 1

Sum of the first n terms of an A.P,

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Given that sum of the first $7$ terms of an A.P is $63$

i. e $S_7=63$

$\Rightarrow \frac{7}{2}[2a+6d]=63$

$\Rightarrow 2a+6d=63\times \frac{2}{7}$

$\Rightarrow 2a+6d=18$----------(1)
Also given sum of its next $7$ terms is $161$.

But Sum of first $14$ terms $=$ sum of first $7$ terms $+$ sum of next $7$ terms.

$S_{14}=63+161=224$

$\Rightarrow \frac{14}{2}[2a+13d]=224$

$\Rightarrow [2a+13d]=224\times \frac{2}{14}$

$\Rightarrow [2a+13d]=32$-------(2)

Solving eq (1) and eq (2) we obtain
$2a+6d-2a-13d=18-32$

$\Rightarrow -7d=-14$

$\Rightarrow d=2$

Now, $2a+6d=18$

$\Rightarrow 2a+6\left(2\right)=18$

$\Rightarrow 2a=6$

$\Rightarrow a=3$

$T_{28}=a+27d$

$=3+27\left(2\right)$

$=3+54$

$=57$

$\therefore T_{28}==57$.

Therefore, ${28}^{th}$ term of this A.P. is $57$..