Let and be the roots of the equation x^2 - 3x - 2 = 0 . If a sequence is defined as S_n = ^n + ^n for n 1 , then the value of the infinite series _ n=1 ^ S_n 4^n is

Step-by-step solution

The given series can be written as: _ n=1 ^ ^n + ^n 4^n = _ n=1 ^ ( 4 )^n + _ n=1 ^ ( 4 )^n From the quadratic equation x^2 - 3x - 2 = 0 , the roots are 3 17 2 . Since 17 Using the sum formula for an infinite GP ( S = a 1-r ): _ n=1 ^ ( 4 )^n = 4 1 - 4 = 4 - Similarly, _ n=1 ^ ( 4 )^n = 4 - Adding the two sums: 4 - + 4 - = (4 - ) + (4 - ) (4 - )(4 - ) = 4( + ) - 2 16 - 4( + ) + From the quadratic equation x^2 - 3x - 2 = 0 , we know that the sum of the roots + = 3 and the product of the roots = -2 . Substituting the