It is the 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, ...
You are watching: Read the numbers and decide what the next number should be. 6 18 20 10 30 32 16
Can you number out the next couple of numbers?
Makes A Spiral
When we make squares v those widths, we obtain a pretty spiral:
Do friend see just how the squares fit neatly together?For example 5 and also 8 make 13, 8 and also 13 make 21, and so on.
The Rule
The Fibonacci Sequence deserve to be written as a "Rule" (see Sequences and Series).
First, the terms space numbered native 0 onwards choose 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 ax number 6 is called x6 (which equates to 8).
Example: the 8th ax isthe 7th hatchet plus the 6th term: x8 = x7 + x6 |
So we deserve to write the rule:
The dominion 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 prior to that (n−2)Example: term 9 is calculated like this:
x9= x9−1 + x9−2
= x8 + x7
= 21 + 13
= 34
Golden Ratio
And here is a surprise. When we take any two succeeding (one ~ the other) Fibonacci Numbers, their ratio is really close come the gold Ratio "φ" i m sorry is roughly 1.618034...
In fact, the enlarge the pair the Fibonacci Numbers, the closer the approximation. Allow us shot 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... | |
... ... | ... |
We don"t have to start with 2 and 3, right here I randomly chose 192 and also 16 (and gained the succession 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):
A
B
B / A
192
16
16
208
13 | ||
208 224 224 432 | 1.92857143... | |
... ... | ... | |
7408 11984 | 1.61771058... | |
11984 19392 | 1.61815754... | |
... ... | ... |
It takes much longer to get good values, however it reflects that not simply the Fibonacci Sequence deserve to do this!
Using The gold Ratio to calculation Fibonacci Numbers
And even an ext surprising is the we have the right to calculate any kind of Fibonacci Number using the golden Ratio:
xn = φn − (1−φ)n√5
The price comes the end as a whole number, specifically equal come the addition of the previous two terms.
Example: x6
x6 = (1.618034...)6 − (1−1.618034...)6√5
When I offered a calculator top top this (only start the golden Ratio come 6 decimal places) I got the prize 8.00000033 , a an ext accurate calculation would be closer to 8.
Try n=12 and also see what friend get.
You can likewise calculate a Fibonacci Number by multiplying the vault Fibonacci Number through the golden Ratio and then round off (works for numbers above 1):
Example: 8 × φ = 8 × 1.618034... = 12.94427... = 13 (rounded)
Some amazing Things
Here is the Fibonacci sequence again:
n = | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ... |
xn = | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | 377 | 610 | ... |
There is an interesting pattern:
Look at the number x3 = 2. Every 3rd number is a many of 2 (2, 8, 34, 144, 610, ...)Look in ~ the number x4 = 3. Every 4th number is a many of 3 (3, 21, 144, ...)Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)And so on (every nth number is a multiple of xn).
Notice the first couple of digits (0,1,1,2,3,5) space the Fibonacci sequence?
In a method they all are, except multiple number numbers (13, 21, etc) overlap, prefer this:
Terms listed below Zero
The sequence works below zero also, 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 the each number is found by including up the two numbers prior to it!)
In reality the sequence below zero has the very same numbers as the sequence over zero, except they monitor a +-+- ... Pattern. It deserve to be written favor this:
x−n = (−1)n+1 xn
Which claims that ax "−n" is same to (−1)n+1 times term "n", and also the value (−1)n+1 nicely makes the correct +1, −1, +1, −1, ... Pattern.
History
Fibonacci was not the an initial to know around the sequence, that was recognized in India hundreds of years before!
About Fibonacci The Man
His genuine name to be Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.
"Fibonacci" was his nickname, which roughly method "Son of Bonacci".
See more: Mood Rings Colors What Do They Mean, Color Chart Meanings
As well together being renowned for the Fibonacci Sequence, he assisted spread Hindu-Arabic numerals (like our current numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) with Europe in location of roman inn Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! give thanks to you Leonardo.
Fibonacci Day
Fibonacci work is November 23rd, as it has actually the number "1, 1, 2, 3" which is component of the sequence. So next Nov 23 let everyone know!