The first two numbers in the fibonacci numbers will be 0 and 1 ($F_{0}$ = 0 and $F_{1}$ = 1) and each subsequent number will be the sum of previous two. It was invented by Leonardo di Pisa and is known as fibonacci.

Applications of fibonacci numbers can be found in fibonacci search technique, fibonacci cubes where it helps in interconnecting parallel and distributed systems. The sequence of fibonacci numbers can be found all over mathematics and also in nature.

The fibonacci numbers are the sequence of numbers ${(F_{n})}_{n = 1}^{\infty}$ defined by the recurrence relation

$F_{n} = F_{n - 1} + F_{n - 2}$
where $F_{0} = 0, F_{1} = F_{2}$ = 1

For any given number now you can generate fibonacci series.
$F_{n}$ is called nth fibonacci number and n > 2.
1. The sum of first n fibonacci numbers is
$u_{1} + u_{2} + ..... + u_{n - 1} + u_{n}$ = $u_{n+2}$ - 1

Proof: By using definition of the fibonacci sequence we have

$u_{1}$ = $u_{3} - u_{2}$

$u_{2}$ = $u_{4} - u_{3}$
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$u_{n - 1}$ = $u_{n+1} - u_{n}$

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

Add all the equations above

$u_{1} + u_{2} + .......+ u_{n-1} + u_{n}$ = $u_{n+2} - u_{2}$

As $u_{2}$ = 1 the above equation leads to

$u_{1} + u_{2} + .......+ u_{n-1} + u_{n}$ = $u_{n+2}$ - 1
Hence the proof.


2. Sum of odd terms of the fibonacci series

$u_{1} + u_{3} + u_{5} + u_{2n - 1} = u_{2n}$

Proof: Looking at the individual terms and by definition we have

$u_{1} = u_{2}$

$u_{3} = u_{4} - u_{2}$

$u_{5} = u_{6} - u_{4}$

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$u_{2n - 1} = u_{2n} - u_{2n-2}$

Adding all the above equations will give us

$u_{1} + u_{3} + u_{5} + u_{2n - 1} = u_{2n}$
Hence proved.
Fibonacci series is the sequence of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21,.....The next number can be easily found by adding two previous numbers.
It is customary to use $F_{0}$ = 0, '0' is the Zeroth term, $F_{1}$ = 1 = $F_{2}$

To find 3rd term (2) in the above sequence add (1 + 1).

To find 4th term (3) in the above sequence add (1 + 2).

To find 5th term(5) in the above sequence add (2 + 3) and so on!

The ratio of two numbers in a fibonacci series will be close to the Golden ratio "$\varphi$" which is appromiately 1.6180...

For example: $\frac{13}{8}$ = 1.625
Given below is the first 21 fibonacci sequence number list.
$F_{ 0 }$
0
$F_{1}$ 1
$F_ {2}$ 1
$F_{3}$ 2
$F_{4}$ 3
$F_{5}$ 5
$F_{6}$ 8
$F_{7}$ 13
$F_{8}$ 21
$F_{9}$ 34
$F_{10}$ 55
$F_{11}$ 89
$F_{12}$ 144
$F_{13}$ 233
$F_{14}$ 377
$F_{15}$ 610
$F_{16}$
987
$F_{17}$
1597
$F_{18}$ 2584
$F_{19}$ 4181
$F_{20}$ 6765


Solved Examples

Question 1: Generate the fibonacci series for first ten numbers.
Solution:
 
The formula to find the fibonacci series is given as

$F_{n}$ = $F_{n-1} + F_{n-2}$  n >2 
We know that $F_{1}$  = $F_{2}$ = 1

Substituting  n = 3 in the above formula we get $F_{3}$ = 2
Substituting  n = 4 in the above formula we get $F_{4}$ = 3
Substituting  n = 5 in the above formula we get $F_{5}$ = 5
Substituting  n = 6 in the above formula we get $F_{6}$ = 8

Continue the same process till n = 10.
Therefore the final sequence  will be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
 

Question 2: What will be the 12th value in fibonacci series
Solution:
 
The formula to find the fibonacci series is given as

$F_{n}$ = $F_{n-1} + F_{n-2}$  n > 2 
To find the 12th value in the fibonacci series we need to the know the values till the 11th term.
As we know $F_{1}$  = $F_{2}$ = 1

Substitute n=3  in the above formula we get $F_{3}$ = 2
Continuing this process till n = 12, we get the sequence as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144

Therefore the 12th term in the fibonacci series is 144.