Question

For positive integers n₁, n₂ the value of the expression

$(1 + i)^{n_{1}} + \left( 1 + i^{3} \right)^{n_{1}} + \left( 1 + i^{5} \right)^{n_{2}} + \left( 1 + i^{7} \right)^{n_{2}}$, where i = $\sqrt{- 1}$, is a real number if and only if

• A. n₁ = n₂ + 1
• B. n₁ = n₂
• C. n₁ = n₂ – 1
• D. All values of n₁ and n₂

Answer 1

Answer: (D)

All values of n₁ and n₂

Answer 2

Sol.$(1 + i)^{n_{1}} + \left( 1 + i^{3} \right)^{n_{1}} + \left( 1 + i^{5} \right)^{n_{2}} + \left( 1 + i^{7} \right)^{n_{2}}$= $(1 + i)^{n_{1}} + (1 - i)^{n_{1}} + (1 + i)^{n_{2}} + (1 - i)^{n_{2}}$= real number for all n₁ and n₂