Let and be the roots of the quadratic equation x^2 - 5x - 2 = 0 . If P_n = ^n + ^n for all natural numbers n , then the value of the summation _ r=0 ⁴ ⁴C_ r 2^ 4-r 5^r P_r is equal to

Step-by-step solution

The given summation is _ r=0 ⁴ ⁴C_ r 2^ 4-r 5^r P_r . Substituting P_r = ^r + ^r , we get: _ r=0 ⁴ ⁴C_ r 2^ 4-r 5^r ( ^r + ^r) This can be separated into two sums: _ r=0 ⁴ ⁴C_ r 2^ 4-r (5 )^r + _ r=0 ⁴ ⁴C_ r 2^ 4-r (5 )^r Recognizing these as binomial expansions, we can write them as: (2 + 5 )^4 + (2 + 5 )^4 Since and are roots of x^2 - 5x - 2 = 0 , we have: ^2 - 5 - 2 = 0 5 + 2 = ^2 ^2 - 5 - 2 = 0 5 + 2 = ^2 Substituting these relations back into the binomial expressions: ( ^2)^4 + ( ^2)^4 = ^8 + ^8 By the definit