Question

If the ratio of the sum of the first n
terms of two A.Ps is (7n + 1) : (4n +
27), then find the ratio of their 9th
terms.​

Answer 1

Let first AP be a1,a2,a3...

Let first term be A and common difference be D.

Sum of n terms= n/2(2A+(n-1)D)

Second AP be b1, b2, b3,...

Let first term be b and common difference be d.

Sum of n terms= n/2(2b +(n-1)d)

sum of n terms of first AP = 2A+(n-1)D

sum of n terms of second AP 2b+(n-1)d

n/2 and n/2 cancel out above

2A+(n-1)D = 7n+ 1

2b+(n-1)d 4n+27

Replace n by 2n-1,

So that it becomes,

2A+(2n-1-1)D

=> 2(A+(n-1)D)

similarly,

=>2(b+(n-1)d)

(A+(n-1)D) = 7(2n-1)+ 1

(b+(n-1)d) 4(2n-1)+27

The required ratio can be found by putting n=9

on RHS.

7(17) + 1

4(17)+27

=> 120

95

=====> that is 24

15

Ratio of their 9th terms is 24/15.