The sum of the first two terms of a G.P. is −4 and the fifth term is 4 times the third term. Then, the first term of the G.P. is :

1. $-\dfrac{3}{4}$ or 4

2. $\dfrac{3}{4}$ or $\dfrac{1}{4}$

3. $\dfrac{4}{3}$ or $\dfrac{1}{4}$

4. $-\dfrac{4}{3}$ or 4

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

Given,

Sum of the first two terms is -4

⇒ a + ar = -4

⇒ a(1 + r) = -4 …..(1)

Given,

The fifth term is 4 times the third term.

⇒ ar^{5 - 1} = 4ar^3-1

⇒ ar⁴ = 4ar²

⇒ r² = 4

⇒ r = $\sqrt{4}$

⇒ r = 2 or r = -2

Substituting r = 2 into equation 1 :

⇒ a(1 + 2) = -4

⇒ 3a = -4

⇒ a = $-\dfrac{4}{3}$.

Substituting r = -2 into equation 1:

⇒ a[1 + (-2)] = -4

⇒ a(1 - 2) = -4

⇒ a(-1) = -4

⇒ a = 4.

Hence, option 4 is the correct option.