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

Let us take terms of GP as $$a,ar,ar^2$$

then the terms in AP will be $$ar,ar^2,a$$

In that case, $$ar^2=\frac{(a+ar)}{2}$$

$$2ar^2=a+ar$$

On simplifying the equation, we get r =  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

$$\frac{a}{1-r\ }=36$$

plugging r=-1/2 we get

a = 54.