(6) Find a close formula of ƒ, using the results of the diagonalization above. Must show all work!

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Fibonacci Numbers:**

\[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots\]

Define the Fibonacci sequence \(\{f_n\}_{n=0}^\infty\) by a recurrence relation (2nd order linear difference equation):

\[f_{n+2} = f_{n+1} + f_n, \quad n \geq 0, \quad f_0 = 0, \quad f_1 = 1.\]

**Fibonacci matrix:**

\[ F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}. \]
Transcribed Image Text:**Fibonacci Numbers:** \[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots\] Define the Fibonacci sequence \(\{f_n\}_{n=0}^\infty\) by a recurrence relation (2nd order linear difference equation): \[f_{n+2} = f_{n+1} + f_n, \quad n \geq 0, \quad f_0 = 0, \quad f_1 = 1.\] **Fibonacci matrix:** \[ F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}. \]
(6) Find a closed formula for \( f_n \), using the results of the diagonalization above. Must show all work!
Transcribed Image Text:(6) Find a closed formula for \( f_n \), using the results of the diagonalization above. Must show all work!
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