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Fibonacci Numbers

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 Sub Topics 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. To 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. 1st element: 0 2nd element: 1 3rd element: 0 + 1 = 1 4th element: 1 + 1 = 2 5th element: 2 + 1 = 3 6th element: 3 + 2 = 5 7th element: 5 + 3 = 8 8th element: 8 + 5 = 13 9th element: 13 + 8 = 21The fibonacci series looks like 0, 1, 1, 2, 3, 5, 8, 13, 21, …. and goes till infinity.

What are Fibonacci Numbers?

Fibonacci number is a sequence of numbers, where each successive number is the sum of the two previous numbers. The sequence of fibonacci numbers is
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 Number Sequence

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, nth 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.

Fibonacci Numbers List

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

Fibonacci Numbers Examples

Given below are some of the examples on fibonacci numbers.

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.

Question 2: Find the value of the 21st 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 21st value in the fibonacci series, we need to know the values till the 20th 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 21st term in the fibonacci series is 10496.