Question

Let a₁, a₂, ..... an be positive real numbers such that a₁a₂a₃ ...... a_n = 1. Then (1 + a₁) (1 + a₂) ..... (1 + a_n) is

(a) > 2 

(b) > 2^n–1 

(c) > 2ⁿ 

(d) < 2^n–1

Answer 1

(c) > 2ⁿ 

∴ (1 + a₁) (1 + a₂) ...... (1 + a_n) > \(2\sqrt{a_1}\times2\sqrt{a_2}\times.....2\sqrt{a_n}\)

⇒ (1 + a₁) (1 + a₂) .... (1 + a_n) > 2ⁿ \(\sqrt{a_1.a_2...a_n}\)

= 2ⁿ \(\sqrt1=2^n.\)