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 found by adding the two numbers before it (1+2),
And the 5 is (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, ...
First, the terms are numbered from 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 ...
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)
Example: term 9 would be calculated like this:
x9 = x9-1 + x9-2 = x8 + x7 = 21 + 13 = 34
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
my brain hurts ._.