If the ratio of the sums of first of two A.P.' s is (7n + 1) : (4n+27), find the ratio of their $m^{t h}$ terms.

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Solution

$\frac{(S n)_{1}}{(S n)_{2}} = \frac{\frac{n}{2} [2 a_{1} + (n - 1) d_{1}]}{\frac{n}{2} [2 a_{2} + (n - 1) d_{2}]} = \frac{7 n + 1}{4 n + 27}$
$= \frac{a_{1} + (\frac{n - 1}{2}) d_{1}}{a_{2} + (\frac{n - 1}{2}) d_{2}} = \frac{7 n + 1}{4 n + 27}$
$\frac{n - 1}{2} = m - 1$
$\Rightarrow n = 2 m - 2 + 1$
$\Rightarrow n = 2 m - 1$
$= \frac{a_{1} + (m - 1) d_{1}}{a_{2} + (m - 1) d_{2}} = \frac{7 (2 m - 1) + 1}{4 (2 m - 1) + 27}$
$\Rightarrow \frac{(T m)_{1}}{(T m)_{2}} = \frac{14 m - 6}{8 m + 23}$ .

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

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