Question

If a_i > 0 ∀ i ∈ N such that ∏ a_i i ∈ [i = 1, n] = 1  then prove that (1 + a₁) (1 + a₂) (1 + a₃) ........(1 + a_n) ≥ 2ⁿ

Answer 1

Using A.M. ≥ G.

=> (1 + a₁) (1 + a₂) (1 + a₃) ........(1 + a_n) ≥ 2ⁿ (a₁a₂a₃.....a_n)^1/n 

As a₁ a₂ a₃ ..... a_n = 1 

Hence (1 + a₁) (1 + a₂) (1 + a₃) ........(1 + a_n) ≥ 2ⁿ