Q.

If the second, third and first terms of a geometric progression (GP) form an arithmetic progression (AP), then find the first term of the GP, given that the sum to infinite terms of the GP is 36.

A. 9

B. 54

C. 27

D. 18

Think of common properties of an AP.

B. 54

Let us take terms of GP as \(\Rightarrow a, ar, ^\)

then the terms in AP will be \(\Rightarrow ar, ^, a\)

In that case, \(^ = \frac\)

\(^=a+ar\)

on simplifying the quadratic we get r = 1  OR  r = -1/2

r can not be 1 as that will make each term of the GP same which will not make it an infinite GP, so we get r = -1/2

⇒ Also given that sum to infinite terms of the GP = 36

\(\frac=36\)

plugging r=-1/2 we get

a = 54.