The sum of 17 terms of two series in A.P. are in the ratio (3n + 8) : (7n + 15). Find the ratio of their 12th terms.

There are two AP with different first term and common difference.

For the First AP

Let first term be a

Common difference = d

Sum of n terms =

and n^th term = a_n = a + (n – 1)d

For the second AP

Let first term be A

Common difference = D

Sum of n terms =

and n^th term = A_n = A + (n – 1)D

It is given that

…(i)

Now, we need to find ratio of their 12^th term

Hence,

n – 1 = 11 × 2

⇒ n = 22 + 1

⇒ n = 23

Putting n = 23 in eq. (i), we get

Hence the ratio of 12^th term of 1^st AP and 12^th term if 2^nd AP is 7:16