**Fibonacci series**is a sequence of numbers where each number is the sum of the two preceding ones. The sequence starts with 0 and 1, and continues with the next term being the sum of the previous two terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

The Fibonacci series was introduced by the Italian mathematician Leonardo Fibonacci in the early 13th century. He discovered the sequence while studying the growth of rabbit populations.

The sequence has many interesting mathematical properties and relationships, and appears in a wide variety of contexts throughout science, nature, and human art and design. For example, the spiral patterns found in sea shells, galaxies, and hurricanes are often described using the Fibonacci sequence and its related phenomena, the Golden ratio and the Golden spiral.

The Fibonacci sequence is not just a mathematical curiosity, but also a practical tool for solving many real-world problems. For example, it can be used in financial modeling for calculating interest rates and investment returns, and in computer algorithms for generating random numbers and searching for data.

In programming, the Fibonacci sequence is often used as an illustrative example for different programming concepts and techniques, due to its simplicity and elegance. The sequence can be generated using loops, recursion, or mathematical formulas, and can help teach concepts such as iterative algorithms, recursive functions, and time complexity analysis.

Overall, the Fibonacci sequence is a widely recognized and celebrated mathematical concept that has captivated the minds of mathematicians, scientists, and artists for centuries, and continues to inspire new discoveries and innovations today.

In C++, Fibonacci series is a sequence of numbers where each number is the sum of the two preceding ones.

**Method 1: Using loops**

In this method, you can generate a Fibonacci sequence by iterating a loop and adding the previous two numbers to generate the next number. Here's an example code that demonstrates generating a Fibonacci sequence using a loop:

`#include <iostream> using namespace std; int main() { int n = 10; // number of terms to generate int a = 0, b = 1, c; // initialize the first two terms cout << "Fibonacci series: "; for (int i = 0; i < n; ++i) { cout << a << " "; // output the current term c = a + b; // calculate the next term a = b; // update the first term b = c; // update the second term } return 0; }`

In this code, the integer variable

`n`

is set to the number of terms to generate. The first two terms of the sequence are initialized as 0 and 1 in variables `a`

and `b`

. A loop is then used to generate and output each subsequent number of the sequence by adding the previous two terms, and updating values of `a`

and `b`

so that the next two terms can be calculated in the next iteration.**Method 2: Using recursion**

In this method, you can generate a Fibonacci sequence recursively by defining a function that calls itself to generate the subsequent terms of the sequence.

`#include <iostream> using namespace std; int fib(int n) { if (n <= 1) { return n; // base case } return fib(n - 1) + fib(n - 2); // recursive call } int main() { int n = 10; // number of terms to generate cout << "Fibonacci series: "; for (int i = 0; i < n; ++i) { cout << fib(i) << " "; // output the current term } return 0; }`

In this code, the integer variable

`n`

is set to the number of terms to generate. The `fib`

function is defined to generate the Fibonacci sequence recursively by calling itself with smaller values of `n`

, which leads to the base case. The base case of the recursive function is when `n`

is 0 or 1, in which case the function simply returns `n`

.The

`main`

function then iterates over each value of `i`

between 0 and `n`

and outputs the corresponding term of the sequence generated by `fib(i)`

.**Method 3: Using Binet's formula**

In this method, you can generate a Fibonacci sequence using Binet's formula, which computes the nth term of the sequence by a mathematical formula.

`#include <iostream> #include <cmath> // required for sqrt using namespace std; int fib(int n) { double phi = (1 + sqrt(5)) / 2; // Golden ratio return round(pow(phi, n) / sqrt(5)); // Binet's formula } int main() { int n = 10; // number of terms to generate cout << "Fibonacci series: "; for (int i = 0; i < n; ++i) { cout << fib(i) << " "; // output the current term } return 0; }`

In this code, the integer variable

`n`

is set to the number of terms to generate. The `fib`

function calculates the `n`

th term of the sequence using Binet's formula, which involves the mathematical constants of the Golden ratio and the square root of 5.The

`main`

function then iterates over each value of `i`

between 0 and `n`

and outputs the corresponding term of the sequence generated by `fib(i)`

.These are different methods to generate a Fibonacci sequence in C++. All these methods work for any number of terms to generate.

**PRACTICE PROBLEMS :**