Let A_r= (x+ 1 x )^3 (x^2+ 1 x^2 )^3 (x^3+ 1 x^3 )^3 . (x^r+ 1 x^r )^3 . If x^2+x+1=0 , then 1 A₃ + 1 A₆ + 1 A₉ + 1 A₁₂ + . . =

Step-by-step solution

It is given that, if x^2+x+1=0 , then A_r= (x+ 1 x )^3 (x^2+ 1 x^2 )^3 (x^3+ 1 x^3 )^3 (x^r+ 1 x^r )^3 As, we know that w and w^2 are roots of quadratic equation x^2+x+1=0 , so w^3=1 and 1+w+w^2=0 A_r= (w+ 1 w )^3 (w^2+ 1 w^2 )^3 (w^3+ 1 w^3 )^3 (w^r+ 1 w^r )^3 A₃=(-1)(-1)(8)=8 Similarly, A₆=8^2, A₉=8^3, and so on. 1 A₃ + 1 A₆ + 1 A₉ + 1 A₁₂ + = 1 8 + 1 8^2 + 1 8^3 + 1 8^4 + = 1 8 1- 1 8 = 1 7