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Fibbonaci Identity. (1 Viewer)

Thread starter Fus Ro Dah
Start date Oct 1, 2012

Fus Ro Dah

Member

#1

$\\ $The Fibbonaci Sequence $ f_1, f_2,..., f_n $ is defined by $ f_1 = f_2 = 1 $ and $ f_n = f_{n-1} + f_{n-2}, n \geq 3. \\\\ $Let $ Q = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}.$ Prove that $ Q^n = \begin{bmatrix} f_{n+1} & f_n \\ f_n & f_{n-1} \end{bmatrix} $ and hence prove that $ f_{3n} = f_{n+1}^3 + f_n^3 - f_{n-1}^3$

A

AAEldar

Premium Member

#2

Did this last session in Linear Algebra. Was fun and interesting to see a different way for the sequence to be done!

Carrotsticks

Retired

#3

1. Induction.

2. Equate the entries of Q^{3n} with the entries of (Q^n)^3.

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