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The sum of n terms of two A Ps in the ratio $2 n + 3 : 6 n + 5$ then the ratio of their $13^{th}$ terms is

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$53 : 155$

$27 : 77$

$29 : 83$

$31 : 89$

answer is A.

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Detailed Solution

$S_{n}^{1} : S_{n}^{2} = 2 n + 3 : 6 n + 5 \Rightarrow \frac{n}{2} [2 a_{1} + (n - 1) d_{1}] : \frac{n}{2} [2 a_{2} + (n - 1) d_{2}] = 2 n + 3 : 6 n + 5$

$2 a_{1} + (n - 1) d_{1} : 2 a_{2} + (n - 1) d_{2} = 5 + (n - 1) 2 : 11 + (n - 1) 6$

$a_{1} = 5 / 2, a_{2} = 11 / 2, d_{1} = 2, d_{2} = 6$

$T_{1}^{1} : T_{2}^{2} = 53 : 155$

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