Question

The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Answer 1

Let a and r be the first term and the common ratio of the G.P. respectively.
∴ a = 1
a3 = ar2 = r 2
a5 = ar4 = r4
∴ r2 + r 4 = 90
⇒ r 4 + r2 - 90 = 0
$⇒ r2^ = \frac{- 1 + \sqrt{1 + 360}}{ 2} = \frac{- 1 ± \sqrt{361}}{ 2} = \frac{- 1 ±19 }{ 2} = -10 \,or\, 9$
∴ r = ±3 (Taking real roots)
Thus, the common ratio of the G.P. is ±3.