the ratio of the sum of n terms of two AP's is (7n+1):(4n+27).find the ratio of their m th terms.
Answer: let a1 , a2 be the 1st terms and d1 , d2 the common differences of the two given A.P's. then the sums of their n terms are given by
Sn = n/2 {2.a1+(n-1)d1} and Sn' = n/2{2. a2 +(n-1)d2}
Sn/Sn' = n/2{2.a1+(n-1)d1} / n/2{2.a2 + (n-1)d2}
Sn / Sn' = 2.a1+(n-1)d1 / 2.a2 + (n-1)d2
it is given that
Sn / Sn' = 7n+1 / 4n + 27
2.a1 + (n-1)d1 / 2.a2 + (n-1)d2 = 7n+1 / 4n+27 ........................ (i)
To find the ratio of the mth terms of the two given AP's , we replace n by (2m-1) in equation (i)
therefore, 2.a1 + (n-1)d1 / 2.a2 + (n-1)d2 = 7(2m-1) + 1 / 4(2m-1) + 27
a1 + (m-1)d1 / a2 + (m-1)d2 = 14m - 6 / 8m + 23
Hence, the ratio of the mth terms of the two A.P's is (14m - 6) : (8m + 23)
My question is, why it has been assumed (2m-1) in the place of 'n' ? has it been arrived from solving with the help of a formula or is it a mere assumption? if it is simply an assumption, why it should be assumed as (2m-1) ? why not 2m or (m - 1)
Here is the proper solution of the question:
In the question, we need to find out the ratio between the mth terms of the two AP's.
And we know that mth term of the the AP = a + (m – 1)d.
So, observe that in (1), if somehow we get this type of expression, then we will easily get the mth term. This will be the case only when we put 
, i.e., n = 2m – 1.
Hope will get the concept now.