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## Homework Statement

Using [tex]u_k = \[ \left( \begin{array}{ccc} F_{k+1} \\ F_k \end{array} \right)\] [/tex] [tex] u_0 = \[ \left( \begin{array}{ccc} 1 \\ 0 \end{array} \right)\][/tex] [tex] A = \[ \left( \begin{array}{ccc} 1 & 1 \\ 1 & 0 \end{array} \right)\][/tex]

Solve for [tex]u_k[/tex] in terms of [tex]u_0[/tex] to show that:

[tex]F_k = \frac{1}{\sqrt{5}}\ \left(\left(\frac{1 + \sqrt{5}}{2}\ \right)^k - \left(\frac{1 - \sqrt{5}}{2}\ \right)^k\right) [/tex]

## Homework Equations

See above.

## The Attempt at a Solution

Well, I worked out that [tex]u_k = A^k u_0 [/tex]

[tex] \[ \left( \begin{array}{ccc} F_{k+1} \\ F_k \end{array} \right)\][/tex] = [tex]\[ \left( \begin{array}{ccc} 1 & 1 \\ 1 & 0 \end{array} \right)\]^k[/tex] [tex]\[ \left( \begin{array}{ccc} 1 \\ 0 \end{array} \right)\] [/tex]

But I'm not sure of the matrix operations I need to use to expand that A matrix, and the other two, in order to obtain an algebraic expression.

Note: That 1,1,1,0 matrix is meant to be to the power of k, but the LaTex went weird.