We come across many series, when we work on numbers like, series of even numbers, odd numbers, prime numbers, series of squares of the numbers and the series of the cubes of the numbers. These are some of the common series we look around us. In this page, you will learn about fibonacci numbers.
1To form the Fibonacci number, we will start from the two smallest whole numbers i.e. 0 and 1. Now, the next element of the Fibonacci number will be calculated using these two numbers. We write the first two elements of the given series and then to find the next element, we write 0 + 1 = 1. So, the series of three elements will be 0, 1, 1. This process continues and below are some of fibonacci numbers. ^{st} element: 0 2 ^{nd} element: 13 ^{rd} element: 0 + 1 = 14 ^{th} element: 1 + 1 = 25 ^{th} element: 2 + 1 = 36 ^{th} element: 3 + 2 = 57 ^{th} element: 5 + 3 = 88 ^{th} element: 8 + 5 = 139 ^{th} element: 13 + 8 = 21The fibonacci series looks like 0, 1, 1, 2, 3, 5, 8, 13, 21, …. and goes till infinity. |

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,........

In mathematical terms, the recurrence relation of fibonacci numbers is

$F_{n}$ = $F_{n - 1}$ + $F_{n - 2}$

where, $F_{0}$ = 0 and $F_{1}$ = 1 Fibonacci numbers have an interesting property. That is, the ratio of two consecutive numbers tends to the golden ratio as the number gets bigger and bigger. It has a value of approximately 1.618034 called the golden section or the golden mean or golden number. It is represented by the letter phi ($\phi$).

Coincidentally, there are 1.609 kilometers in a mile, which is within 0.5% of the Golden ratio. As the ratio is also almost the same as kilometers per mile,

The above can be written as

$\frac{F_{n + 1}}{F_{n}}$ = $\frac{[Mile]}{[Km]}$

$\rightarrow$ $F_{n}$.[Mile] = $F_{n + 1}$.[Km]

Which means, n

^{th}fibonacci number in miles is the same as (n + 1)

^{th}fibonacci number in kilometers.

Any fibonacci number can be calculated using the golden ratio.

$x_{n}$ = $\frac{\varphi^{n} - (1 - \varphi)^{n}}{\sqrt{5}}$

where, $\varphi$ = 1.6180. → Read More Given below are the first 30 fibonacci sequence number list:

n |
F(n) (Factorisation) |

0 | 0 |

1 | 1 |

2 | 2 |

3 | 3 |

4 | 5 |

5 | 8 |

6 | 8 |

7 | 13 |

8 | 21 |

9 | 34 |

10 | 55 |

11 | 89 |

12 | 144 |

13 | 243 |

14 | 377 |

15 | 610 |

16 | 987 |

17 | 1597 |

18 | 2584 |

19 | 4181 |

20 | 6765 |

21 | 10946 |

22 | 17711 |

23 | 28657 |

24 | 46368 |

25 | 75025 |

26 | 121393 |

27 | 196418 |

28 | 317811 |

29 | 514229 |

30 | 832040 |

### Solved Examples

**Question 1:**Find the fibonacci series for the first five numbers.

**Solution:**

The formula to find the fibonacci series is given as

$F_{n}$ = $F_{n - 1}$ + $F_{n - 2}$, where n > 2

We know that $F_{1}$ = $F_{2}$ = 1

We need to find the remaining three terms.

Plug in n = 3 in the formula

$F_{3}$ = $F_{2}$ + $F_{1}$

= 1 + 1

= 2

Similarly, substituting n = 4 and n = 5 respectively in the formula, we get 3 and 5.

Therefore, the first five numbers in fibonacci sequence are 1, 1, 2, 3, 5.

$F_{n}$ = $F_{n - 1}$ + $F_{n - 2}$, where n > 2

We know that $F_{1}$ = $F_{2}$ = 1

We need to find the remaining three terms.

Plug in n = 3 in the formula

$F_{3}$ = $F_{2}$ + $F_{1}$

= 1 + 1

= 2

Similarly, substituting n = 4 and n = 5 respectively in the formula, we get 3 and 5.

Therefore, the first five numbers in fibonacci sequence are 1, 1, 2, 3, 5.

**Question 2:**Find the value of the 21

^{st}number in fibonacci series.

**Solution:**

The formula to find the fibonacci series is given as

$F_{n}$ = $F_{n - 1}$ + $F_{n - 2}$, where n > 2

We know that $F_{1}$ = $F_{2}$ = 1

To find the 21

Substituting n = 3 in the above formula, we get F$_{3}$ = 2

Continuing this process till n = 21, we get the sequence as 1, 1, 2, 3, 5, 8,13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946.

The 21

$F_{n}$ = $F_{n - 1}$ + $F_{n - 2}$, where n > 2

We know that $F_{1}$ = $F_{2}$ = 1

To find the 21

^{st}value in the fibonacci series, we need to know the values till the 20^{th}term.Substituting n = 3 in the above formula, we get F$_{3}$ = 2

Continuing this process till n = 21, we get the sequence as 1, 1, 2, 3, 5, 8,13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946.

The 21

^{st}term in the fibonacci series is 10496.