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Question

Let $a$ be the frist term and $d$ be the common difference of given $A.P.$

Second term,$a+d=12$       .....$(1)$

Sum of fris

Vedclass · Accepted Answer

$20$

Vedclass · Answer

Let $a$ be the frist term and $d$ be the common difference of given $A.P.$

Second term,$a+d=12$       .....$(1)$

Sum of frist nine terms,

${S_9} = \frac{9}{2}\left( {2a + 8d} ight) = 9\left( {a + 4d} ight)$

Given that ${S_9}$ is more than $200$ and less than $200$

$ \Rightarrow 200 < {S_9} < 220$

$ \Rightarrow 200 < 9\left( {a + 4d} ight) < 220$

$ \Rightarrow 200 < 9\left( {a + d + 3d} ight) < 220$

Putting value of $(a+d)$ from equation $(1)$

$200 < 9\left( {12 + 3d} ight) < 220$

$ \Rightarrow 200 < 108 + 27d < 220$

$ \Rightarrow 200 - 108 < 108 + 27d - 108 < 220 - 108$

$ \Rightarrow 92 < 27d < 112$

Possible value of $d$ is $4$

$27 	imes 4 = 108$

Thus, $92<108<112$

Putting value of $d$ in equation $(1)$

$a+d=12$

$a=12-4=8$

${4^{th}}$ term $ = a + 3d = 8 + 3 	imes 4 = 20$

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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