Let be a root of the equation x^2+x+1=0 . The number of values of n 1, 2, 3, , 50 such that (1+ )^n = 1 + ^n is

Step-by-step solution

The roots of x^2+x+1=0 are and ^2 . Let = . The given equation is (1+ )^n = 1 + ^n . Since 1+ = - ^2 , the left hand side is (- ^2)^n = (-1)^n ^ 2n . Let us check the validity of the equation for n = 1, 2, 3, 4, 5, 6 to find the pattern: For n=1 : LHS = - ^2 . RHS = 1+ = - ^2 . (Satisfied) For n=2 : LHS = (-1)^2 ^4 = . RHS = 1+ ^2 = - . (Not satisfied) For n=3 : LHS = (-1)^3 ^6 = -1 . RHS = 1+ ^3 = 2 . (Not satisfied) For n=4 : LHS = (-1)^4 ^8 = ^2 . RHS = 1+ ^4 = 1+ = - ^2 . (Not satisfied) For n=5 : LHS = (-1)^5