Definition
The Fibonacci sequence is the sequence defined by \(F_n = F_{n-1} + F_{n-2}\) with \(F_1 = 1\) and \(F_2 = 1\).
Exercise
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Write out the first 10 terms of the Fibonacci sequence.
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Using the facts that the 21st term of the sequence is 10946 and the 23rd term of the sequence is 28657, determine the 22nd term.
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Show that the Fibonacci sequence is “almost” geometric in the sense that you multiply by approximately the same number to move from one term to the next. What would you estimate this “approximate” common ratio to be?
Definition
The golden ratio, denoted by \(\phi\), is the number \(\phi = \displaystyle\frac{1+\sqrt{5}}{2} = 1.61803\ldots\).
Exercise
We noticed above that the Fibonacci sequence is “approximately” geometric with an “approximate” common ratio close to the golden ratio.
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Let’s look at the sequence \(\frac{\phi}{\sqrt{5}}, \frac{\phi^2}{\sqrt{5}}, \frac{\phi^3}{\sqrt{5}},\ldots\). Notice that this is a geometric series with first term \(\frac{\phi}{\sqrt{5}}\) and common ratio $\phi$. Compute the value of the first 8 terms, and fill in the table below. What do you notice?
\(\frac{\phi}{\sqrt{5}}\) \(\frac{\phi^2}{\sqrt{5}}\) \(\frac{\phi^3}{\sqrt{5}}\) \(\frac{\phi^4}{\sqrt{5}}\) \(\frac{\phi^5}{\sqrt{5}}\) \(\frac{\phi^6}{\sqrt{5}}\) \(\frac{\phi^7}{\sqrt{5}}\) \(\frac{\phi^8}{\sqrt{5}}\) Value: -
Use what you observed above to complete the statement of Binet’s formula below.
Binet’s formula for the Fibonacci sequence
Let \(\phi\) be the golden ratio. A formula for the $n$th term of the Fibonacci sequence is
$F_n = $
Exercise
Use Binet’s formula to compute \(F_{40}\).
Exercise
The Fibonacci sequence shows up in many places in nature, art, and elsewhere. Do a quick search, and write down three of the places it can be found that you find most interesting.