Question

If A = 1 + r^a + r^2a + ⋯ ∞, then express r in terms of ′a′ and ′A′.

Answer 1

Given series is in G.P. with common ratio r^a

∴ Sum of infinite terms of G.P. = \(\frac{x}{1-r}\)

∴ A = \(\frac{1}{1-r^a}\)                    [x = 1]

⇒ A (1 − r^a) = 1

⇒ A − Ar^a = 1

⇒ A − 1 = Ar^a

⇒ r^a = \(\frac{A-1}{A}\)

⇒ r = \(\big(\frac{A-1}{A}\big)^\frac{1}{a}\)