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

Given: S₂ = —4 and a₅ = 4 × a₃

The Sum of terms in G.P. is given by:

∴ S₂ = = —4

⇒ = —4

⇒ a(1+r) = —4 –1

Here,

a₅ = 4 × a₃

⇒ ar⁴ = 4 × ar² (∵ nth term of the G.P is ar^n-1)

⇒ r² = 4

⇒ r = 2

∴ r = +2 or –2

Case 1: r = +2

From eq –1

a(1+r) = —4

⇒ a(1+2 ) = —4

⇒ a =

∴ G.P :

Case 2: r = –2

From eq –1

a(1+r) = —4

⇒ a(1+(–2) ) = —4

⇒ a = = 4

∴ G.P : 4 , –8, 16, –32,…….

∴ The possible G.P ‘s are or 4 , –8, 16, –32,…….