The limiting sum of the GP is twice the sum of the first two terms. Find the possible values of the common ratio?

Let the first term of the GP be a and the common ratio be r.

The sum of the first two terms of the GP is a + ar = a(1 + r).

The limiting sum of the GP is given by S = a/(1 - r) when |r| < 1.

According to the given condition, the limiting sum is twice the sum of the first two terms:

2(a + ar) = a/(1 - r)

2a + 2ar = a/(1 - r)

2a + 2ar = a(1 + r)

2 + 2r = 1 + r

r = -1

Therefore, the only possible value of the common ratio is -1.