Question

The sum of the first $7$ terms of an A.P is $63$ and the sum of its next $7$ terms is $161$. Find the $28$th term of this A.P.

Answer 1

Sum of first $n$ terms of an $A.P.=\frac{n}{2}[2a+(n-1\left)d\right]$

Given:

$S_7=63$

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

$2a+6d=18$ ...... (1)

Also, sum of next $7$ terms is $161$. Thus,

Sum of first $14$ terms $=$ Sum of first $7$ terms $+$ Sum of next $7$ terms

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

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

$2a+13d=32$ ...... (2)

Solving equations (1) and (2), we get,

$a=3$ and $d=2$

Therefore,

$T_{28}=a+(28-1)d$

$T_{28}=3+27\times 2$

$T_{28}=3+54=57$

Hence, ${28}^{th}$ term of the $A.P.$ is $57$.