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What Is the Fibonacci Sequence and Why is it Famous?

Learn what the Fibonacci sequence is, its relationship to population growth, and how it can make you the life of the party.

By
Jason Marshall, PhD
6-minute read
Episode #16

The Fibonacci Sequence and Rabbits

Fibonacci’s Rabbits

No, not really. To explain, let’s head way back to the year 1202, at which point we meet our hero: Fibonacci—a bright young Italian guy from Pisa in his 20s who’d done a lot of traveling. After returning to Italy, and inspired by what he learned about math during his travels, Fibonacci wrote a book. But this wasn’t just any book—this book turned out to be...well, important. For one thing, Fibonacci used it to introduce Europe to the 0 through 9 numeral system we still use today. Without that, who knows—we might all be stuck counting with Roman numerals!

Among other things, Fibonacci’s book also included a musing about a math problem which turned out to have a far more interesting and lasting solution than anyone could have imagined. Fibonacci’s math question seems simple enough: If two newborn rabbits are placed in a pen, how many rabbits will the pen contain after one year? To answer his question, Fibonacci wanted us to assume the following:

  1. whenever a pair of rabbits reproduces, they always produce one male and one female offspring;

  2. rabbits can reproduce once per month;

  3. rabbits can start to reproduce when they are one month old; and

  4. rabbits never die.

So that’s the question. What’s the solution?

The Fibonacci Sequence

To start answering the question, let’s think in terms of how many pairs of rabbits there are at the beginning of each month. Start with the 1 newborn pair that exists at the beginning of the first month. These first two newborns are too young to reproduce that month, so we begin the second month with 1 pair as well. So far the sequence is

1, 1.

Not very exciting, but let’s keep going. At the beginning of the second month, the original pair is mature enough to mate. As a result, one new pair of rabbits is born at the end of the second month. So at the beginning of the third moth, we have a total of 2 pairs. The original pair again mates at the beginning of that month, but the newborn pair is still immature. The original pair produces another pair of offspring, so at the beginning of the fourth month, we have a total of 3 pairs of rabbits. The sequence is now

1, 1, 2, 3,

which is a little more interesting, but still fairly mundane.

However, now things start to get exciting...and potentially confusing too—so stick with me. In fact, if you’re finding this a little hard to follow, check out the Math Dude’s “Video Extra!” for episode 16 on YouTube for a more graphical explanation. But getting back to our story... At the beginning of the fourth month, two pairs mate (the original, and the first pair of offspring), and one pair is still immature. Those two pairs that mated each produce a new pair, giving us 5 pairs at the beginning of the fifth month. Let’s go through one more month. At the beginning of the fifth month, three pairs mate, but the newest two pairs that were just born are still immature. After the three new pairs of offspring are born, our total moves to 8 pairs.

At this point, the sequence is:

1, 1, 2, 3, 5, 8.

Do you see a pattern? Would it help if I said the next number is 13? And the next after that is 21?

1, 1, 2, 3, 5, 8, 13, 21, …

I’ll admit, the pattern isn’t totally obvious at first. But after you see it, it is. The trick is that each number in the Fibonacci sequence is obtained by adding together the previous two:

  • 1 + 1 = 2 is the third number,

  • 1 + 2 = 3 is the fourth number,

  • 2 + 3 = 5 is the fifth number,

  • 3 + 5 = 8 is the sixth number,

  • 5 + 8 = 13 is the seventh number,

  • 8 + 13 = 21 is the eighth number, and so on.

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About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.

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