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: Sequence & Series (Mathematics)

$200 < a_1 + a_2 + a_3 + \cdots + a_9 < 220\\$ or $200 < \frac{9}{2}\,[2a_1 + 8d] < 220$ $\\ \Rightarrow 200 < 9a_1 + 36d < 220 ....(1)\\$ Since $a_2 = 12$ $[a_1 + d = 12$ $\\ \Rightarrow d = 12 - a_1\\$ Substituting in equation (1), $200 < 9a_1 + 36(12 - a_1) < 220$ $\\ \Rightarrow 200 < 9a_1 + 432 - 36a_1 < 220$ $\\ \Rightarrow 200 < 432 - 27a_1 < 220$ $\\ \Rightarrow -232 > 27a_1 > 212$ $\\ \Rightarrow a_1 = 8$ $\\ \Rightarrow d = 12 - 8 = 4$ $\\ \Rightarrow a_3 = a_1 + 2d = 16, a_4 = a_1 + 3d = 20$

Explanation

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