Question

An AP consists of $23$ terms. If the sum of the $3$ terms in the middle is $141$ and the sum of the last $3$ terms is $261$, then the first term is

• A. $6$
• B. $5$
• C. $4$
• D. $3$
• E. $2$

Answer 1

Answer: (B)

$5$

Let the first term and common difference of AP be $a$ and $d$, respectively.
Given, $a_{11}+a_{12}+a_{13}=141$
$\Rightarrow 3a+33d=141$
$\Rightarrow a+11d=47$ ....(i)
Also given, $a_{21}+a_{22}+a_{23}=261$
$\Rightarrow 3a+63d=261$
$\Rightarrow a+21d=87$ .....(ii)
Subtract equations (i) and (ii), we get

$10d=40$

$\Rightarrow d=4$

Putting this value in equation (i), we get

$a+44=47$

$\Rightarrow a=47-44=3$

Therefore, first term is $3$.