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 10th terms.

Answer 1

Given proportion of whole of n terms of two AP's = (7n+1):(4n+27) We should consider the proportion these two AP's mth terms as am : a'm →(2) Review the nth term of AP equation, a = a + (n – 1)d Henceforth condition (2) moves toward becoming, am : a'm = a + (m – 1)d : a' + (m – 1)d' On duplicating by 2, we get am : a'm = [2a + 2(m – 1)d] : [2a' + 2(m – 1)d'] = [2a + {(2m – 1) – 1}d] : [2a' + {(2m – 1) – 1}d'] = S2m – 1 : S'2m – 1 = [7(2m – 1) + 1] : [4(2m – 1) +27] [from (1)] = [14m – 7 +1] : [8m – 4 + 27] = [14m – 6] : [8m + 23] In this manner the proportion of mth terms of two AP's is [14m – 6] : [8m + 23].