<s>[INST] Closed form of the Fibonacci sequence: solving using the characteristic root method Here is the official theorem I'll use: 
Since the Fibonacci sequence is defined as $F_n=F_{n-1}+F_{n-2}$, we solve the equation $x^2-x-1=0$ to find that $r_1 = \frac{1+\sqrt 5}{2}$ and $r_2 = \frac{1-\sqrt 5}{2}$
So we have $F_n = c_1\left(\frac{1+\sqrt 5}{2}\right)^n + c_2\left(\frac{1-\sqrt 5}{2}\right)^n$
We know that $F_0 = F_1 = 1$. So we can solve the following system to find the values of $c_1$ and $c_2$: 
$1 = c_1 + c_2$
$1 =  c_1\left(\frac{1+\sqrt 5}{2}\right) + c_2\left(\frac{1-\sqrt 5}{2}\right)$
Solving this system does not give $c_1 = 1/\sqrt5, c_2 = -1/\sqrt 5$ , even though that is apparently the right answer, i.e. the closed form of the Fibonacci sequence is apparently  $$\frac1{\sqrt 5}\left(\frac{1+\sqrt 5}{2}\right) -\frac1{\sqrt 5}\left(\frac{1-\sqrt 5}{2}\right)$$ 
Where did I go wrong? Why doesn't solving the system of equations give me $c_1 = 1/\sqrt5, c_2 = -1/\sqrt 5$?
 [/INST] Let's see... 
$$f_n = \left\{
\begin{array}{ll}
0 & \text{ for } n = 0 \\
1 & \text{ for } n = 1 \\
f_{n-1} + f_{n-2} & \text{ for } n>1
\end{array}
\right.$$
Now, the recursion can be written as $$f_n - f_{n-1} - f_{n-2} = 0,$$ so characteristic equation is $$x^2-x-1=0.$$
Now, the roots of the equation are  $$X_{1,2} = \frac{1 \pm \sqrt{5}}2,$$
so general solution is
$$f_n = C_1\cdot\left(\frac{1 + \sqrt{5}}2\right)^n + C_2\left(\frac{1 - \sqrt{5}}2\right)^n$$
From the $f_1$ and $f_2$ we get
\begin{eqnarray}
0 &=& C_1 + C_2 \\
1 &=& C_1\left(\frac{1 + \sqrt{5}}2\right) + C_2\left(\frac{1 - \sqrt{5}}2\right)
\end{eqnarray}
From the first equation we get $$C_2 = -C_1,$$ so
\begin{equation}
1 = C_1\left(\frac{1 + \sqrt{5}}2\right) -C_1\left(\frac{1 - \sqrt{5}}2\right)
\end{equation}
Now, we have $$C_1\left[\frac{1 + \sqrt{5}}2 - \frac{1 - \sqrt{5}}2\right] = 1$$ or
$$C_1\cdot\sqrt{5}  =1$$
So, $$C_1 = \frac{1}{\sqrt{5}}.$$
Now, $$C_2 = -\frac{1}{\sqrt{5}}.$$ The particular solution for the equation is therefore
$$f_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}2\right)^n - \left(\frac{1-\sqrt{5}}2\right)^n\right]$$
</s>