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      <text>8.4.6 (Sequences in nature)</text>
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      <text><![CDATA[<h1>Sequences in nature</h1>
<p>In this section we will return to a topic that we touched upon briefly, in a different setting, namely the Fibonacci sequence. We mentioned in the previous section that the Fibonacci sequence was developed to answer a thought experiment that concerned the breeding of rabbits. In this section we will detail the statement and solution of this problem. Before this however, we should note that the Fibonacci sequence appears throughout nature. We have seen in week one, that the Fibonacci sequence is governed by <em>the golden ratio</em>, and many things grow or are formed according to this ratio. The figure below recalls some examples. However, it is not only the Fibonacci sequence that can be found in nature. One major example is Di Vinci's Vitruvian man, this shows the mathematical proportion of the human body. Similarly, we have seen snail shells grow according to the Fibonacci sequence. Indeed anything that has a repeating pattern is often goverened by mathematical sequences, for example, think of honeycomb in beehives. </p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Fibonacci%20Flower.jpg" alt="Fibonacci spirals" width="538" height="343" /></p>
<p> </p>
<p>We will now consider the thought experiment of the multiplying rabbits.</p>
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<h3>Example</h3>

<p>Beginning with a single pair of rabbits, how many pairs of rabbits will be born in a year, assuming that every month each male and female rabbit gives birth to a new pair of rabbits, and the new pair of rabbits itself starts giving birth to additional pairs of rabbits after the first month of their birth?
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<h3>Solution</h3>

<p>The way Fibonacci thought about this was to count the number of rabbits after each month. As the rabbits take one month to mature, they take one month before a new pari of rabbits can breed. We count the pairs.</p>
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<p><b>month 1:</b> There are the initial pair of rabbit. They are not mature so they cannot reproduce. So the total number of pairs the only \(1\)-the initial pair.</p>
<p><b>month 2:</b> In month two the initial pair of rabbits cannot yet reproduce. therefore we still only have the initial pair. </p>
<p><b>month 3:</b>At month three, the initial pari of rabbits have reached maturity and can therefore reproduce. So at the end of this month we have the initial pair as well as their offspring.</p>
<p><b>month 4:</b>Now, the initial pair is still mature so they reproduce. In addition to the offspring from the previous month this gives us 3 pairs.</p>
<p><b>month 5:</b>In this month, the initial rabbit reproduce, as do their first offspring. In addition we have the rabbits born in the previous month which are not mature, this gives a total of 5 pairs of rabbits. </p>
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<p>We can see that the total number of pairs of rabbits at the end of each month go 
\[
1,1,2,3,5,\dots.
\]
In other words they follow the Fibonacci sequence, where the next entry in the sequence is the sum of the previous two.</p>]]></text>
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