Let \(A_0 = 6 \\ A_1 = 5 \\ A_n = A_{n - 1} + A_{n - 2} \text{for } n \geq 2\).

There is a unique ordered pair (c,d) such that \(c\phi^n + d\widehat{\phi}^n\) is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences.

\(\phi = \frac{1 + \sqrt{5}}{2} \; \text{and} \; \widehat{\phi} = \frac{1 - \sqrt{5}}{2}\)

A0=6, A1 = 5, A2=11, A3=16, A4 = 27, A5 = 43...

Im not seeing any patterns

pls help

I found this https://math.stackexchange.com/questions/2215103/lucas-and-fibonacci-numbers but i don't get the final solution (whats c?)

Guest Dec 14, 2020

edited by
Guest
Dec 14, 2020