It is that simple!
Here is a much 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: What is the next number 12 13 15
Can you figure out the next few numbers?
Makes A Spiral
When we make squares v those widths, we acquire a pretty spiral:
Do friend see exactly how the squares fit nicely together?For example 5 and also 8 do 13, 8 and also 13 make 21, and also so on.
The Rule
The Fibonacci Sequence can be created as a "Rule" (see Sequences and also Series).
First, the terms are numbered indigenous 0 onwards prefer 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 hatchet number 6 is dubbed x6 (which equals 8).
Example: the 8th hatchet isthe 7th ax plus the 6th term: x8 = x7 + x6 |
So we have the right to write the rule:
The dominance is xn = xn−1 + xn−2
where:
xn is hatchet number "n"xn−1 is the previous hatchet (n−1)xn−2 is the term prior to that (n−2)Example: term 9 is calculated favor this:
x9= x9−1 + x9−2
= x8 + x7
= 21 + 13
= 34
Golden Ratio
And below is a surprise. When we take any type of two succeeding (one ~ the other) Fibonacci Numbers, their proportion is very close to the golden Ratio "φ" i m sorry is approximately 1.618034...
In fact, the larger the pair that 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... | |
... ... | ... |
We don"t need to start through 2 and also 3, right here I randomly chose 192 and also 16 (and got the sequence 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 shows that not simply the Fibonacci Sequence can do this!
Using The gold Ratio to calculate Fibonacci Numbers
And even much more surprising is the we can calculate any Fibonacci Number using the golden Ratio:
xn = φn − (1−φ)n√5
The prize comes out as a whole number, specifically equal come the enhancement 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 beginning the golden Ratio to 6 decimal places) I acquired the answer 8.00000033 , a an ext accurate calculation would certainly be closer come 8.
Try n=12 and see what girlfriend get.
You can likewise calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the gold Ratio and also then round off (works because that numbers over 1):
Example: 8 × φ = 8 × 1.618034... = 12.94427... = 13 (rounded)
Some interesting Things
Here is the Fibonacci succession 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 exciting pattern:
Look in ~ 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 multiple of 3 (3, 21, 144, ...)Look in ~ the number x5 = 5. Every 5th number is a many of 5 (5, 55, 610, ...)And for this reason on (every nth number is a multiple of xn).
Notice the first few digits (0,1,1,2,3,5) space the Fibonacci sequence?
In a way they all are, except multiple number numbers (13, 21, etc) overlap, prefer this:
Terms below Zero
The succession works listed 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 come yourself the each number is discovered by adding up the 2 numbers before it!)
In reality the sequence below zero has the exact same numbers together the sequence over zero, other than they follow a +-+- ... Pattern. It can be written favor this:
x−n = (−1)n+1 xn
Which states that term "−n" is same to (−1)n+1 times term "n", and also the worth (−1)n+1 nicely makes the correct +1, −1, +1, −1, ... Pattern.
History
Fibonacci was no the an initial to know around the sequence, it was recognized in India hundreds of years before!
About Fibonacci The Man
His genuine name was Leonardo Pisano Bogollo, and he lived in between 1170 and 1250 in Italy.
"Fibonacci" to be his nickname, i m sorry roughly way "Son the Bonacci".
See more: What Are The Subunits Of Dna Is Composed Of Repeating Subunits Called ?
As well together being well known for the Fibonacci Sequence, he assisted spread Hindu-Arabic numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) v Europe in place of roman inn Numerals (I, II, III, IV, V, etc). That has saved us all a most trouble! give thanks to you Leonardo.
Fibonacci Day
Fibonacci day is November 23rd, together it has actually the digits "1, 1, 2, 3" i m sorry is component of the sequence. So following Nov 23 let everyone know!