Let z = 1 - i 3 2 , i = - 1 . Then the value of 21 + z + 1 z 3 + z 2 + 1 z 2 3 + z 3 + 1 z 3 3 + … + z 21 + 1 z 21 3 is______.

Step-by-step solution

Given: z = 1 - i 3 2 ⇒ z = - - 1 + i 3 2 ⇒ z = − ω where, ω is the cube root of unity. So, 1 + ω + ω 2 = 0 ω 3 = 1 Now, let A = z + 1 z 3 + z 2 + 1 z 2 3 + z 3 + 1 z 3 3 + … + z 21 + 1 z 21 3 ⇒ A = − ω − 1 ω 3 + ω 2 + 1 ω 2 3 + − ω 3 − 1 ω 3 3 + … . + − ω 21 − 1 ω 21 3 ⇒ A = - ω 2 + 1 ω 3 + ω 4 + 1 ω 2 3 + − 1 − 1 1 3 + … . + − &#9