### What is the Fibonacci series?

The Fibonacci sequence is a sequence where the next term is the sum of the previous two terms. The first two terms of the Fibonacci sequence are 0 followed by 1.

```
The Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21 ....
```

In mathematical terms,the sequence Fn of Fibonacci numbers is defined by recursion relation

```
Fn = Fn-1 + Fn-2
```

where

```
F0 = 0 , F1 = 1
```

### We will discuss three ways to write the Fibonacci series program :

- Fibonacci series using recursion
- Fibonacci series using Iterative method
- Fibonacci series using Matrix Exponentiation

### (1) Recursive Method :

A simple method that is a direct recursive implementation mathematical recurrence relation is given above.

```
//pseudo code
RFib(n)
{
if n=0 return 0;
else if n=1 return 1;
else return (RFib(n-1)+RFib(n-2));
}
```

*Time Complexity: T(n) = T(n-1) + T(n-2) which is exponential.
Extra Space: O(n) if we consider the function call stack size, otherwise O(1).*

### (2) Iterative Method :

We can optimize the space used in Recursive Method by storing the previous two numbers only because that is all we need to get the next Fibonacci number in series.

```
//pseudo code
IFib(n)
if n=0 return 0;
else if n=1 return 1;
else { a <= 0; b<=1;
for(i=2 to n) do
{ temp <= b;
b<= a+b;
a <= temp;
}
return b;
```

*Time complexity: O(n)
Extra space: O(1)*

### (3) Matrix Exponentiation :

Let n > 1

*Time complexity : O(log(n))
Extra space : O(log(n)) if we consider the function call stack size, otherwise O(1).*

Thanks for reading

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