Let α and β be the roots of x 2 - x - 1 = 0 , with α > β . For all positive integers n , define a n = α n - β n α - β ,   n ≥ 1 b 1 = 1 and b n = a n - 1 + a n + 1 ,   n ≥ 2 . Then which of the following options is/are correct?

Step-by-step solution

Given, α and β are roots of x 2 - x - 1 =   0 a k + 2 - a k = α k + 2 - β k + 2 - α k - β k α - β = α k + 2 - α k - β k + 2 - β k α - β = α k α 2 - 1 - β k β 2 - 1 α - β = a k + 1 ⇒ a k + 2 - a k = a k + 1   ⇒ a k + 2 - a k + 1 = a k ∑ a k k = 1 n = a k + 2 - a 2   = a k + 2 - α 2 - β 2 α - β = a k + 2 - α + β = a k + 2 - 1 ⇒ a 1 + a 2 + a 3 + . . .