Let n denote the number of solutions of the equation z 2 + 3 z ¯ = 0 , where z is a complex number. Then the value of ∑ k = 0 ∞ 1 n k is equal to

Step-by-step solution

Given z 2 + 3 z ¯ = 0 Put z = x + i y , then we know that z ¯ = x - i y , hence, we have x + i y 2 + 3 x - i y = 0 ⇒   x 2 + i 2 y 2 + 2 i x y + 3 x - 3 i y = 0 We also, know that i 2 = - 1 , hence, we get ⇒   x 2 - y 2 + 2 i x y + 3 x - 3 i y = 0 ⇒   x 2 - y 2 + 3 x + i ( 2 x y - 3 y ) = 0 + i 0 On comparing the real and imaginary parts, we get x 2 - y 2 + 3 x = 0         … 1 And 2 x y - 3 y = 0       … 2 ⇒   y 2 x