Question

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than $200$ and less than $220$. If the second term in it is $12$, then $4^{th}$ term is :

• A. $8$
• B. $16$
• C. $20$
• D. $24$

Answer 1

The correct option is C $20$

Let ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{d}^{\prime }$ be the first term and common difference respectively.
Sum of the first $9$ terms= $\frac{9[2a+8d]}{2}$ = $9(a+4d)$
$a+d=12$
$a+3d=x$
$\frac{220}{9}>(a+4d)>\frac{200}{9}$
$\frac{220}{9}>a+3d+d>\frac{200}{9}$
$\frac{220}{9}>x+d>\frac{200}{9}$
$d=\frac{x-12}{2}$
$\frac{220}{9}>x+\frac{x-12}{2}>\frac{200}{9}$
$\frac{440}{9}>3x-12>\frac{400}{9}$
$\frac{548}{27}>x>\frac{508}{27}$
Only one option satisfies the given range i.e.$x=20$