Find a G.P. for which the sum of first two terms is -4 and the fifth term is 4 times the third term.

Given,

⇒ a + a₂ = -4

⇒ a + ar = -4

⇒ a(1 + r) = -4 ……..(i)

⇒ a₅ = 4a₃

⇒ ar⁴ = 4ar²

⇒ r² = 4

⇒ r = ± 2

Let r = 2.

Substituting value of r in (i) we get,

⇒ a(1 + 2) = -4

⇒ 3a = -4

⇒ a = $-\dfrac{4}{3}$.

G.P. = a, ar, ar², ……..

= $-\dfrac{4}{3}, -\dfrac{8}{3}, -\dfrac{16}{3}$……..

Let r = -2.

Substituting value of r in (i) we get,

⇒ a(1 + (-2)) = -4

⇒ -a = -4

⇒ a = 4.

G.P. = a, ar, ar², ……..

= 4, -8, 16, -32,……..

Hence, G.P. = $-\dfrac{4}{3}, -\dfrac{8}{3}, -\dfrac{16}{3}$…….. or 4, -8, 16, -32.