If is a root of the equation x^2+x+1=0 , then the value of _ r=0 ¹⁰ ¹⁰C_ r ( ^r + ^ 2r )^2 is equal to

Step-by-step solution

The roots of x^2+x+1=0 are the complex cube roots of unity, and ^2 . Let = . The general term in the sum is T_r = ¹⁰C_ r ( ^r + ^ 2r )^2 . Case 1: If r is a multiple of 3, then ^r = 1 and ^ 2r = 1 . ( ^r + ^ 2r )^2 = (1+1)^2 = 4 . Case 2: If r is not a multiple of 3, then ^r + ^ 2r = + ^2 = -1 . ( ^r + ^ 2r )^2 = (-1)^2 = 1 . Thus, we can write ( ^r + ^ 2r )^2 = 1 + 3 , where = 1 if r is a multiple of 3, and 0 otherwise. The required sum is: S = _ r=0 ¹⁰ ¹⁰C_ r (1) + 3 _ r=0, 3|r ¹⁰ ¹⁰C_ r We know that _ r=0 ¹⁰ ¹⁰C