#### Question

If the ratio of the sum of first n terms of two A.P’s is (7n +1): (4n + 27), find the ratio of their mth terms.

#### Solution

Let a_{1}, a_{2} be the first terms and d1, d2 the common differences of the two given A.P’s.

Then we have `S_n=n/2[2a_1+(n-1)d_1] " and " S_n=n/2[2a_2+(n-1)d_2]`

∴ `S_n/S_n=(n/2[2a_1+(n-1)d_1])/(n/2[2a_2+(n-1)d_2])=(2a_1+(n-1)d_1) /(2a_2+(n-1)d_2)`

It is given that `S_n/S_n=(7n+1)/(4n+27)`

`:.(2a_1+(n-1)d_1)/(2a_2+(n-1)d_2)=(7n+1)/(4n+27) "....(1)"`

To find the ratio of the mth terms of the two given A.P.'s replace n by (2m-1) in equation (1).

`:.(2a_1+(2m-1-1)d_1)/(2a_2+(2m-1-1)d_2)=(7(2m-1)+1)/(4(2m-1)+27)`

`:.(2a_1+(2m-1)d_1)/(2a_2+(2m-2)d_2)=(14m-7+1)/(8m-4+27)`

`:.(a_1+(m-1)d_1)/(a_2+(m-1)d_2)=(14m-6)/(8m+23)`

Hence, the ratio of the mth terms of the two A.P's is 14m-6 : 8m + 23