(Fibonacci numbers) The sequence Fn of Fibonacci numbers is defined by the recurrence relation: Fn = Fn-1 + Fn-2, n > 3,

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Fibonacci Numbers The Sequence Fn Of Fibonacci Numbers Is Defined By The Recurrence Relation Fn Fn 1 Fn 2 N 3 1 (165.83 KiB) Viewed 29 times

(Fibonacci numbers) The sequence Fn of Fibonacci numbers is defined by the recurrence relation: Fn = Fn-1 + Fn-2, n > 3, = (1) with F1 = F2 = 1. To find a closed-form expression for Fn, we can proceed as follows. First, we form a sequence of 2-dimensional column vectors as 圆圆 (2) By eq. (1), we have Fit1 F: [:] - [ ] E] 1 i=1,2,..., Fi+2 2 (3) li 1 will give the term right to it. i.e., left multiplying any term in (2) by the matrix Denoting and Xk = [Fk+1 A= - [1 ] 2 k = 0,1, 2..., Fk+2] 2 then eq. (3) is converted into the difference equation Xk+1 = Axk with Xo = 0 (4) Therefore, finding a closed-form expression for Fn is equivalent to find to a solution of (4). Multiplying the first term n – 2 times by A will result in Xn-2 = F Fn-1 n-2 - [4- () - [7 10 * [H = ' . = An2 = = Xn-2 = Fn - An-2xo. Hence, to find a solution of (4), it is essential to obtain a formula for Ak. Find An-2 and then derive a formula of the closed-form expression for Fr.