 # Fibonacci Sequence – Math is Fun The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

The next number is found by adding up the two numbers before it:

• the 2 is found by adding the two numbers before it (1+1),
• the 3 is found by adding the two numbers before it (1+2),
• the 5 is (2+3),
• and so on!

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,377,610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, …

Can you figure out the next few numbers?

## Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together? For example 5 and 8 make 13, 8 and 13 make 21, and so on.

This spiral is found in nature! See: Nature, The Golden Ratio, and Fibonacci

## The Rule

The Fibonacci Sequence can be written as a “Rule” (see Sequences and Series).

First, the terms are numbered from 0 onwards like this:

So term number 6 is called x6 (which equals 8).

So we can write the rule:

The Rule is xn = xn−1 + xn−2

where:

• xn is term number “n”
• xn−1 is the previous term (n−1)
• xn−2 is the term before that (n−2)

## Golden Ratio

And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio “φ” which is approximately 1.618034…

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In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

1.5 1.666666666… 1.6 1.625 … 1.618055556… 1.618025751… …

We don’t have to start with 2 and 3, here I randomly chose 192 and 16 (and got the sequence 192, 16,208,224,432,656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, …):

0.08333333… 13 1.07692308… 1.92857143… … 1.61771058… 1.61815754… …

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

## Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

xn = φn − (1−φ)n√5

The answer comes out as a whole number, exactly equal to the addition of the previous two terms.

You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):

## Terms Below Zero

The sequence works below zero also, like this:

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- … pattern. It can be written like this:

x−n = (−1)n+1 xn

Which says that term “−n” is equal to (−1)n+1 times term “n”, and the value (−1)n+1 neatly makes the correct +1, −1, +1, −1, … pattern.

## History

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

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