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  4. 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 its 4th term is :

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Q. 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 its $4^{th}$ term is :

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Solution:

$\left(12 - d\right) + 12 + \left(12 + d\right) + \left(12 + 2d\right) +.... 12 +7d$
$=12 × 9 + 27d = 108 + 27d$
now according to question
$200 < 108 + 27d < 220$
$92 < 27d < 112$
$\frac{92}{27} < d < \frac{112}{27} $
$\Rightarrow d = 4$ only integer
$\Rightarrow 4$th term = $12 + 2d = 12 + 8 = 20$