(50g) Nth Fibonacci Number

[Thomas Klemm]

(02-26-2015 01:42 PM)rprosperi Wrote:  So, can you explain that?

For \(x_{n+2}=x_n+x_{n+1}\) you can define a 2nd sequence \(y_n=x_{n+1}\).
Thus the relation becomes:

\(y_{n+1}=x_{n+2}=x_n+y_n\).

Now you combine both sequences into a vector:

\(z_n=\begin{bmatrix}
x_n \\
y_n
\end{bmatrix}\)

This allows to merge both equations into a single one:

\(z_{n+1}=\begin{bmatrix}
x_{n+1} \\
y_{n+1}
\end{bmatrix}=\begin{bmatrix}
y_n \\
x_n+y_n
\end{bmatrix}=\begin{bmatrix}
0 & 1 \\
1 & 1
\end{bmatrix}\cdot\begin{bmatrix}
x_n \\
y_n
\end{bmatrix}=M\cdot z_n\)

But this is just the recursive definition of a geometric sequence leading to:

\(z_n=M^n\cdot z_0\) with \(z_0=\begin{bmatrix}
1 \\
0
\end{bmatrix}\)

HTH
Thomas