If a and b are the roots of the equation x 2 - 7 x - 1 = 0 , then the value of a 21 + b 21 + a 17 + b 17 a 19 + b 19 is equal to

Step-by-step solution

Since, a and b are roots of x 2 - 7 x - 1 = 0 , so a 2 - 7 a - 1 = 0 ⇒ a 2 - 1 = 7 a ⇒ a 4 + 1 - 2 a 2 = 49 a 2 ⇒ a 4 + 1 = 51 a 2       . . . 1 Similarly, b 4 + 1 = 51 b 2       . . . 2 Now, a 21 + b 21 + a 17 + b 17 a 19 + b 19 = a 17 a 4 + 1 + b 17 b 4 + 1 a 19 + b 19 = 51 a 19 + 51 b 19 a 19 + b 19 = 51