Fibonacci Sequence
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)
- Similarly, the 3 is just (1+2),
- And the 5 is just (2+3),
- and so on!
Example: the next number in the sequence above would be (21+34) = 55
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, ...
Can you figure out the next few numbers?
The Rule
The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series):
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)
The terms are numbered form 0 onwards like this:
| n = |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
... |
| xn = |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 |
89 |
144 |
233 |
377 |
... |
Example: term 6 would be calculated like this:
x6 = x6-1 + x6-2 = x5 + x4 = 5 + 3 = 8
Golden Ratio
 |
And here is a surprise. If you 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...
In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few: |
|
A |
B |
|
B / A |
|
2 |
3 |
|
1.5 |
|
3 |
5 |
|
1.666666666... |
|
5 |
8 |
|
1.6 |
|
8 |
13 |
|
1.625 |
|
... |
... |
|
... |
|
144 |
233 |
|
1.618055556... |
|
233 |
377 |
|
1.618025751... |
|
... |
... |
|
... |
Note: this also works if you pick two random whole numbers to begin the sequence, such as 192 and 16 (you would get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):
| A |
B |
|
B / A |
192 |
16 |
|
0.08333333... |
16 |
208 |
|
13 |
208 |
224 |
|
1.07692308... |
224 |
432 |
|
1.92857143... |
... |
... |
|
... |
7408 |
11984 |
|
1.61771058... |
11984 |
19392 |
|
1.61815754... |
... |
... |
|
... |
It takes longer to get good values, but it shows you that it is not just the Fibonacci Sequence that can do this!
Using The Golden Ratio to Calculate Fibonacci Numbers
And even more surprising is this formula for calculating any Fibonacci Number using the Golden Ratio:
Amazingly the answer always comes out as a whole number, exactly equal to the addition of the previous two terms.
Example:
When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8.
Try it for yourself!
Terms Below Zero
The sequence can be extended backwards!
Like this:
| n = |
... |
-6 |
-5 |
-4 |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
... |
| xn = |
... |
-8 |
5 |
-3 |
2 |
-1 |
1 |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
... |
(Prove to yourself that adding the previous two terms together still works!)
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.
About Fibonacci The Man
His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
"Fibonacci" was his nickname, which roughly means "Son of Bonacci".
As well as being famous for the Fibonacci Sequence, he helped spread through Europe the use of Hindu-Arabic Numerals (like our present number system 0,1,2,3,4,5,6,7,8,9) to replace Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.
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