If n₁, n₂ are positive integers, then (1+i)^ n₁ + (1+i^3 )^ n₁ + (1+i^5 )^ n₂ + (1+i^7 )^ n₂ is a real number if and only if

Step-by-step solution

Expression can be written as =(1+i)^ n₁ +(1-i)^ n₁ +(1+i)^ n₂ +(1-i)^ n₂ =2^ n₁ / 2 ( 1 2 +i 1 2 )^ n₁ +2^ n₁ / 2 ( 1 2 -i 1 2 )^ n₁ +2^ n₂ / 2 ( 1 2 +i 1 2 )^ n₂ +2^ n₂ / 2 ( 1 2 -i 1 2 )^ n₂ =2^ n₁ / 2 ( 4 +i 4 )^ n₁ + ( 4 -i 4 )^ n₁ +2^ n₂ / 2 ( 4 +i 4 )^ n₂ + ( 4 -i 4 )^ n₂ +2^ n₂ / 2 n₂ 4 +i n₂ 4 . .+ n₂ 4 - n₂ 4 =2^ n₁ / 2 2 n₁ 4 +2^ n₂ / 2 2 n₂ 4 = real ie, n₁, n₂ are any two positive integers.