Question

If the ratio of the sums of first n terms of two A.P.s is `(5n + 13)/(7n + 27)`, then the ratio of their 4^th terms is ______.

Answer 1

If the ratio of the sums of first n terms of two A.P.s is `(5n + 13)/(7n + 27)`, then the ratio of their 4^th terms is 12 : 19.

Explanation:

For an A.P., `S_n = n/2 [2a + (n - 1)d]` 

So `(S_(n1))/(S_(n2)) = [2a_1 + (n - 1)d_1]/[2a_2 + (n - 1)d_2]` 

= `(5n + 13)/(7n + 27)` 

To get the ratio of the m-th terms a + (m – 1)d, replace `(n - 1)/2` by (m – 1), i.e. take n = 2m – 1.

For m = 4, n = 7, so the ratio of 4^th terms is (5 × 7 + 13) : (7 × 7 + 27) = 48 : 76 = 12 : 19.