If x²+x+1=0 , then the value of (x+ 1 x )⁴+ (x²+ 1 x² )⁴+ (x³+ 1 x³ )⁴+ + (x²⁵+ 1 x²⁵ )⁴ is:

Step-by-step solution

Given x^2 + x + 1 = 0 , so x = (primitive cube root of unity) with x^3 = 1 . From the equation: x + 1 x = -1 . Since x^3 = 1 , value of x^n + x^ -n has period 3: n 0 3 : x^n + x^ -n = 2 n 1, 2 3 : x^n + x^ -n = -1 Fourth powers: (2)^4 = 16 , (-1)^4 = 1 . For n = 1 to 25 : 8 multiples of 3 contribute 8 16 = 128 , remaining 17 terms contribute 17 1 = 17 . Total = 128 + 17 = 145 .