Just what are Fibonacci Numbers?

Here is an amazing mathematical recipe for a series of numbers:

    1. Write down any two numbers, one after the other.
    2. Write down the next number in the series as the sum of the first two numbers.
    3. Repeat step 2 many times, ... or continue forever!

Help: Suppose you first wrote down 5 and then 2. The next number would be 5+2 or 7. Then, the next would be 2+7 or 9. The next would be 7+9=16, and so on, forever. Your series could be written as 5, 2, 7, 9, 16, 25, 41, 66, ...

Play with this mathematic progression and see what you can find out about it. Here are some activities to help you warm up to Fibonacci Numbers:

    • If you graph this series, you get an interesting graph.
    • If your first two numbers are 1 and 1, you generate what are called the Fibonacci Numbers: 1, 1, 2, 3, 5, 8, 13, 21,... If you count the number of spirals in pine cones going to the right, and the number going to the left, you usually get two adjacent Fibonacci number.
    • Divide one number by the one before it to get a ratio. As the numbers get larger, something amazing happens to these ratios.
      • Help: Using consecutive Fibonacci numbers, the first ratio would be 2/5 or 0.4 The next would be 7/2 or 3.5. The sequence of ratios would be 0.4, 3.5, 1.285714286, 1.777777778, 1.5625, 1.64,... You may have to continue this process for five or ten more terms to see what amazing thing happens.

 

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