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.

Let a and r be the first term and the common ratio of the G.P. respectively.

∴ a₁ = 1

nth term of G.P is given by -

a_n = ar^{n - 1}

a₃ = ar² = (1)r²

a₆ = ar⁴ = (1)r⁴

According to question -

a₃ + a₆ = 90

⇒ r² + r⁴ = 90

⇒ r⁴ + r² – 90 = 0

Assume r² = t

⇒ t² + t – 90 = 0

⇒ t² + 10t - 9t – 90 = 0

⇒ t(t + 10) - 9(t + 10) = 0

⇒ (t - 9)(t + 10) = 0

∴ t =9 or t = - 10(invalid because t = r², so t can't be - ve.)

∴ r² = 9 ( Taking real roots)

∴ r = 3

Thus, the common ratio of the G.P. is 3.