Question

A famous sequence $\left\{f_{n}\right\}$, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200 , is defined by the recursion formula $$f_{1}=f_{2}=1, \quad f_{n+2}=f_{n+1}+f_{n}$$ (a) Find $f_{3}$ through $f_{10}$. (b) Let $\phi=\frac{1}{2}(1+\sqrt{5}) \approx 1.618034$. The Greeks called this number the golden ratio, claiming that a rectangle whose dimensions were in this ratio was "perfect." It can be shown that $$\begin{aligned} f_{n} &=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] \\ &=\frac{1}{\sqrt{5}}\left[\phi^{n}-(-1)^{n} \phi^{-n}\right] \end{aligned}$$ Check that this gives the right result for $n=1$ and $n=2$. The general result can be proved by induction (it is a nice challenge). More in line with this section, use this explicit formula to prove that $\lim _{n \rightarrow \infty} f_{n+1} / f_{n}=\phi$. (c) Using the limit just proved, show that $\phi$ satisfies the equation $x^{2}-x-1=0$. Then, in another interesting twist, use the Quadratic Formula to show that the two roots of this equation are $\phi$ and $-1 / \phi$, two numbers that occur in the explicit formula for $f_{n}$.

          A famous sequence $\left\{f_{n}\right\}$, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200 , is defined by the recursion formula
$$f_{1}=f_{2}=1, \quad f_{n+2}=f_{n+1}+f_{n}$$
(a) Find $f_{3}$ through $f_{10}$.
(b) Let $\phi=\frac{1}{2}(1+\sqrt{5}) \approx 1.618034$. The Greeks called this number the golden ratio, claiming that a rectangle whose dimensions were in this ratio was "perfect." It can be shown that
$$\begin{aligned}
f_{n} &=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] \\
&=\frac{1}{\sqrt{5}}\left[\phi^{n}-(-1)^{n} \phi^{-n}\right]
\end{aligned}$$
Check that this gives the right result for $n=1$ and $n=2$. The general result can be proved by induction (it is a nice challenge). More in line with this section, use this explicit formula to prove that $\lim _{n \rightarrow \infty} f_{n+1} / f_{n}=\phi$.
(c) Using the limit just proved, show that $\phi$ satisfies the equation $x^{2}-x-1=0$. Then, in another interesting twist, use the Quadratic Formula to show that the two roots of this equation are $\phi$ and $-1 / \phi$, two numbers that occur in the explicit formula for $f_{n}$.

Added by Danielle S.

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