Let α , β α > β be the roots of the quadratic equation x 2 - x - 4 = 0 . If P n = α n - β n , n ∈ ℕ , then P 15 P 16 - P 14 P 16 - P 15 2 + P 14 P 15 P 13 P 14 is equal to _____.

Step-by-step solution

Given, P n = α n - β n and α   &   β are roots of x 2 - x - 4 = 0 Now putting n - 1 in P n = α n - β n we get, P n - 1 = α n - 1 - β n - 1 Now subtracting P n - P n - 1 we get, P n - P n - 1 = α n - β n - α n - 1 - β n - 1 ⇒ P n - P n - 1 = α n - 2 α 2 - α - β n - 2 β 2 - β Now using the equation α 2 - α - 4 = 0   &   β 2 - β - 4 = 0 we get, P n - P n - 1 = 4 α n - 2 -