Sequences

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.1 – Sequences

Sequences

NOTE: You will not be assessed on topics in this application document.

Interesting Facts about the Fibonacci Sequence

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{F_{n+1}}{F_n}=\varphi \approx 1.61803399}$$\displaystyle\lim_{n\to\infty}{\frac{F_{n+1}}{F_n}}=\phi\approx 1.61803399$

Derivation of the above limit:

Given ${\displaystyle F_{n+1}=F_n+F_{n-1}}$$\displaystyle F_{n+1}=F_n+F_{n-1}$.

1. Let ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{F_{n+1}}{F_n}=L}$$\displaystyle \lim_{n\to\infty}{\frac{F_{n+1}}{F_n}}=L$. We need to find the value of $L$$L$.

2. hen ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{F_n+F_{n-1}}{F_n}=L}$$\displaystyle \lim_{n\to\infty}{\frac{F_n+F_{n-1}}{F_n}}=L$. Explain.

3. Then ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}(\frac{F_n}{F_n}+\frac{F_{n-1}}{F_n})=1+\frac{1}{L}}$$\displaystyle \lim_{n\to\infty}\left({\frac{F_n}{F_n}}+{\frac{F_{n-1}}{F_n}}\right)=1+\frac{1}{L}$. Explain.

4. We know ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{F_{n+1}}{F_n}=\underset{n\to \mathrm{\infty }}{lim}(\frac{F_n}{F_n}+\frac{F_{n-1}}{F_n})}$$\displaystyle \lim_{n\to\infty}{\frac{F_{n+1}}{F_n}}=\lim_{n\to\infty}\left({\frac{F_n}{F_n}}+{\frac{F_{n-1}}{F_n}}\right)$. Explain.

5. Therefore, ${\displaystyle L=1+\frac{1}{L}}$$\displaystyle L=1+\frac{1}{L}$. Explain.

6. Solve for $L$$L$:

   • ${\displaystyle L=1+\frac{1}{L}\Rightarrow L^2=L+1\Rightarrow L^2-L-1=0}$$\displaystyle L=1+\frac{1}{L}\Rightarrow L^2=L+1\Rightarrow L^2-L-1=0$

A Use of the Fibonacci Sequence

Activity 13

We know: $1\text{ mile}\approx 1.60934\text{ km}$$1 \text{ mile} \approx 1.60934 \text{ km}$.

1. Explain how to use the Fibonacci sequence to convert miles into approximate kilometers.
2. Explain how to use the Fibonacci sequence to convert kilometers into approximate miles.

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Last Modified: Tuesday, 10 November 2020 6:38 EDT