Let α = - 1 + i 3 2 . If a = 1 + α ∑ k = 0 100 α 2 k and b = ∑ k = 0 100 α 3 k , then a and b , are the roots of the quadratic equation.

Step-by-step solution

Given, α = - 1 + i 3 2 a = 1 + α ∑ k = 0 100 α 2 k and b = ∑ k = 0 100 α 3 k Now α = ω     ;   ω = - 1 + i 3 2 Using this a and b can be written as a = 1 + ω 1 + ω 2 + ω 4 + … ω 198 + ω 200 = 1 + ω 1 - ω 2 101 1 - ω 2 = 1 + ω 1 - ω 1 - ω 2 = 1 Similarly, b = 1 + ω 3 + ω 6 + … + ω 300 = 101 So, the required quadratic equation is x 2 - a + b x + a b = 0 ⇒ x 2 - 102 x +