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 m^th terms. (CBSE 2015)

Given ratio of sum of n terms of two AP’s = (7n+1) : (4n + 27)

Let’s consider the ratio of these two AP’s m^th terms as a_m : A_m -------------(2)

n^th term of AP formula, aⁿ = 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 = [2a + 2(m – 1)d] : [2a’ + 2(m – 1)d’]

= [2a + {(2m – 1) – 1}d] : [2a’ + {(2m – 1) – 1}d’]

= S_{(2m – 1)} : S’_{(2m – 1)}

= [7(2m – 1) + 1] : [4(2m – 1) +27] ------------------[from (1)]

= [14m – 7 + 1] : [8m – 4 + 27]

= [14m – 6] : [8m + 23]

Thus the ratio of m^th terms of two AP’s is [14m – 6] : [8m + 23].