We often give students a program to write – “To generate a Fibonacci series or Fibonacci Numbers” up to a given no. We need to understand what exactly is Fibonacci series? What is its importance in our lives? Is there any or not?
What is Fibonacci Series?
In the field of mathematics – Fibonacci series or Fibonacci numbers is a series of integers that follows some exact sequence or pattern. Here first two numbers are usually 0 and 1 and every subsequent number is the sum of its previous two numbers. First two numbers can also be 1 and 1. For example:
0  1  1  2  3  5  8  13  21  34  55  89 
Observe, first two numbers are 0 and 1, and every subsequent numbers is sum of previous two numbers.
0  1  1 
next
0  1  1  2 
next
0  1  1  2  3 
next
0  1  1  2  3  5 
and so on…
0  1  1  2  3  5  8  13  21  34  55  89 
From where did Fibonacci numbers or series originated from?
It has its links to Indian Mathematics is relation with Sanskrit prosody, though I have also googled up this information, if you are interested you can find more information on wikipedia.
Ok, how about this – I found this puzzle quite interesting on wikipedia:
Assuming :
 A newly born pair of rabbits (one male and one female) are put in a field;
 Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits;
 Rabbits never die and a mating pair always produces one new pair (one male and one female) every month from the second month on.
 The puzzle that Fibonacci posed was: how many pairs will there be in one year?
If you find it bit difficult – you can find the answer on wikipedia.
Fibonacci series is used so many different fields:
 The Fibonacci numbers occur in the sums of “shallow” diagonals in Pascal’s triangle.
 The Fibonacci Sequence, Spirals and the Golden Mean.
 Fibonacci Numbers As A Basis of the Zodiac
 If you divide the Fibonacci numbers by 10
 It results in a sequence of 60 numbers,
 If you placed them evenly around a circle produce the number zero four times precisely at the beginning of the cardinal zodiac signs.
 The number 5 appears at the beginning of the other zodiac signs
 If you divide the Fibonacci numbers by 10
 Fibonacci series in nature
 Find more about fibonacci – Fibonacci for Kids on Pinterest.com
Few useful images related to Fibonacci series (courtesy wikipedia):
 A tiling with squares whose side lengths are successive Fibonacci numbers
 The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling
 The Fibonacci numbers are the sums of the “shallow” diagonals (shown in red) of Pascal’s triangle.
Few useful Links:

Program to generate a Fibonacci series up to a given number using C++?

Program to generate a Fibonacci series up to a given number using Net beans (Java)?