The Fibonacci Sequence Is The Series Of Numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... The Next Number In The Sequence Above Would Be 21+34 = 55

The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... the next number in the sequence above would be 21+34 = 55

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

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 ...

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)



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

If you take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ"

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

..

Every nth number is a multiple of xn

F(n+1) = round( F(n) Phi ) for all n > 1 Example: Since F(4)=3 then F(5) = round( 3 Phi ) = round( 3x1·618... ) = round( 4·854... ) =5

Fib(n) =

𝑃ℎ𝑖 𝑛 −(−𝑝ℎ𝑖)𝑛 𝑃ℎ𝑖−(−𝑝ℎ𝑖)

=

𝑃ℎ𝑖 𝑛 −(−𝑝ℎ𝑖)𝑛 5

=

1 5

1+ 5 2

𝑛

−

1− 5 2

𝑛