Let and be the roots of the equation z^2 - 2z + 2 i = 0 , where i = -1 . If a sequence is defined as P_n = ^n + ^n for all integers n , then the value of Re (w) + Im (w) , where w = P₁₅ + 16 P₇ P₁₁ , is equal to

Step-by-step solution

Given the equation z^2 - 2z + 2 i = 0 , we can rearrange it as: z^2 + 2 i = 2z Dividing by z , we get: z + 2 i z = 2 Squaring both sides: (z + 2 i z )^2 = 4 z^2 + 4 i ^2 z^2 + 2(z) ( 2 i z ) = 4 z^2 - 4 z^2 + 4 i = 4 z^2 - 4 z^2 = 4 - 4 i Squaring both sides again: (z^2 - 4 z^2 )^2 = (4 - 4 i )^2 z^4 + 16 z^4 - 2(z^2) ( 4 z^2 ) = 16 - 32 i + 16 i ^2 z^4 + 16 z^4 - 8 = 16 - 32 i - 16 z^4 + 16 z^4 = 8 - 32 i Since and are roots of the equation, they both satisfy this relation: ^4 + 16 ^4 = 8 - 32 i and ^4 + 16 ^4 = 8