If ratio of sum of first n terms of 2 APs is (7n+1):(4n+27), then find the ratio of their mth term?

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Solution

Given ratio of sum of n terms of two AP's = (7n+1):(4n+27)
Let's consider the ratio these two AP's mth terms as $a_{m} : a_{m}^{'} \to$ (2)
Recall the nth term of AP formula, $a_{n} = a + (n - 1) d$
Hence equation (2) becomes,
$a_{m} : a_{m}^{'} = a + (m - 1) d : a^{'} + (m - 1) d^{'}$
On multiplying by 2, we get
$a_{m} : a_{m}^{'} = [2 a + 2 (m - 1) d] : [2 a^{'} + 2 (m - 1) d^{'}]$
$= [2 a + (2 m - 1) - 1 d] : [2 a^{'} + (2 m - 1) - 1 d^{'}]$
$= S_{2} m - 1 : S_{2}^{'} m - 1$
$= [7 (2 m - 1) + 1] : [4 (2 m - 1) + 27]$ [from (1)]
$= [14 m - 7 + 1] : [8 m - 4 + 27]$
$= [14 m - 6] : [8 m + 23]$
Thus the ratio of mth terms of two AP’s is $[14 m - 6] : [8 m + 23]$ .

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