The ratio of the sum to $n$ terms of two $A . P^{'} s$ is $(7 n + 1) (4 n + 27)$ . Then the ratio of their $m^{t h}$ terms is ?

(14 m + 6) : (8 m - 23)

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(14 + 6) : (8 m + 23)

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(14 m - 6) : (8 m + 23)

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(14 m - 6) : (8 m - 23)

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Solution

The correct option is C $(14 m - 6) : (8 m + 23)$
$\frac{\frac{n}{2} (2 a_{1} + (n - 1) d_{1})}{\frac{n}{2} (2 a_{2} + (n - 1) d_{2})} = \frac{7 n + 1}{4 n + 27}$

$\frac{\frac{n}{2} [2 a_{1} + (m - 1) d_{1}]}{\frac{n}{2} a_{2} + (m - 1) d_{2}} = ?$

$\frac{a_{1} + \frac{(n - 1)}{2} d_{1}}{a_{2} + \frac{(n - 1)}{2} d_{2}} = \frac{4 n + 1}{4 n + 27}$

Putting $\frac{n - 1}{2} \Rightarrow m - 1$

$\frac{a_{1} + (m - 1) d_{1}}{a_{2} + (m - 1) d_{2}} = \frac{7 \times (2 m - 1) + 1}{4 \times (2 m - 1) + 27}$

$\frac{a m_{1}}{a m_{2}} = \frac{14 m - 6}{8 m + 3}$

$\frac{a m_{1}}{a m_{2}} = \frac{14 m - 6}{8 m - 23}$ .

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