Let (F_n) denote the Fibonacci sequence. Define the Lucas sequence L_0 = 2, L_1

Question

Let (F_n) denote the Fibonacci sequence. Define the Lucas sequence L_0 = 2, L_1 = 1, and for n greaterthanorequalto 1 define L_n+1 = L_n + L_n-1. (For some proofs you may want to use the second principle of induction stated in the problem section of by Chapter 18 as Theorem 18.9.) (a) Calculate L_1, ...., L_10. (b) Calculate L_n - F_n - 1, for n greaterthanorequalto 1. Find a remarkable pattern in this list of numbers. State it clearly and prove it by induction (c) Calculate F_n + L_n. Find a remarkable pattern in this list of clearly and prove it using part (b).

Explanation / Answer

Fibonacci -

Fn = Fn-1 + Fn-2 ; F0 = 0, F1 = 1

Lucas -

Ln = Ln-1 + Ln-2 ; L0 = 2, L1 = 1

b)

Lucas - 2, 1, 3, 4, 7, 11, 18, 29, 47

Fibonacci - 0, 1, 1, 2, 03, 05, 08, 13

Ln - Fn-1 -   1, 2, 3, 5, 08, 13, 21

This is another series like Lucas and Fibonacci with seed values as 1 & 2.

c)

Lucas - 2, 1, 3, 4, 7, 11, 18, 29, 47

Fibonacci - 0, 1, 1, 2, 3, 05, 08, 13, 21

Ln + Fn - 2, 2, 4, 6, 10, 16, 26, 42, 68

This is another series like Lucas and Fibonacci with seed values as 2 & 2.

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