How do I show:
Given we let A=:
|1 1|
|1 0|
That:
A^n is equal to the matrix:
|fn+1 fn|
|fn fn-1|
for whenever R is positive.
Hi
Let $\displaystyle A^n = \left(\begin{array}{cc}a_n&b_n\\c_n&d_n\end{array} \right)$
Then $\displaystyle A^{n+1} = A \cdot A^n = \left(\begin{array}{cc}a_n+c_n&b_n+d_n\\a_n&b_n\en d{array}\right)$
Therefore
$\displaystyle a_{n+1} = a_n + c_n$
$\displaystyle b_{n+1} = b_n + d_n$
$\displaystyle c_{n+1} = a_n$
$\displaystyle d_{n+1} = b_n$
The first and third relations give $\displaystyle a_{n+2} = a_{n+1} + c_{n+1} = a_{n+1} + a_n$
The second and fourth relations give $\displaystyle b_{n+2} = b_{n+1} + d_{n+1} = b_{n+1} + b_n$
Check the first terms of $\displaystyle a_n$ and $\displaystyle b_n$ to determine the index of f
The third and fourth relations give the rest of the solution