Question

lf the ratio of the sums of first n terms of two ap is (7n+ 1) ratio (4n + 27) find the ratio of their mth terms????​

Answer 1

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

We can consider the 9th term as the m th term.
Let’s consider the ratio these two AP’s m th terms as

am : a’m →(2)

Recall the nth term of AP formula,
an = a + (n – 1)d

Hence equation (2) becomes,

am : a’m = a + (m – 1)d : a’ + (m – 1)d’

On multiplying 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]

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

hope it helps u