Spivak's Calculus - Fibonacci Sequence In one of the questions in Spivak's Calculus he defines a Fibonacci sequence $a_1$, $a_2$, $a_3$ as:
$$a_1 = 1$$
$$a_2 = 1$$
$$a_n = a_{n-1} + a_{n-2}$$
for $n\ge3$
and proves $$a_n = \frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt{5}}$$
According to his proof:

But from his proof I don't quite understand how he got from the second line to the third and from the third to the fourth. What throws me off here is where he got the $(1+\frac{1+\sqrt5}{2})$ from and the sudden n, $(1+\frac{1-\sqrt5}{n})$ under the last fraction. For someone who understands what Spivak did, could you please explain this to me, thanks.

Going from the second line to the third line, he uses that
\begin{align*}
\left(\frac{1 + \sqrt{5}}{2}\right)^2 &= \frac{1 + 2 \sqrt 5 + 5}{4} \\
&= \frac{4 + (2 + 2 \sqrt 5)}{4} \\
&= 1 + \frac{1 + \sqrt 5}{2}
\end{align*}
Going from line three to four is just using the fact that
$$a^{n - 2} a^2 = a^{n - 2 + 2} = a^n$$
