C Program
#include <stdio.h> int fib(int n){return n<=1?n:fib(n-1)+fib(n-2);} int main(){ int n=5; for(int i=0;i<n;i++) printf("%d ",fib(i)); }
C Output
Input: n = 5 Output: 0 1 1 2 3
C++ Program
#include <bits/stdc++.h> using namespace std; int fib(int n){return n<=1?n:fib(n-1)+fib(n-2);} int main(){ int n=7; for(int i=0;i<n;i++) cout<<fib(i)<<" "; }
C++ Output
Input: n = 7 Output: 0 1 1 2 3 5 8
JAVA Program
class Main{ static int fib(int n){return n<=1?n:fib(n-1)+fib(n-2);} public static void main(String[] args){ int n=6; for(int i=0;i<n;i++) System.out.print(fib(i)+" "); } }
JAVA Output
Input: n = 6 Output: 0 1 1 2 3 5
Python Program
def fib(n): return n if n<=1 else fib(n-1)+fib(n-2) n=8 for i in range(n): print(fib(i), end=" ")
Python Output
Input: n = 8 Output: 0 1 1 2 3 5 8 13
Detailed Explanation
Problem Statement
The Fibonacci Series is a list of numbers in which every term is the sum of the two preceding it, from 0 and 1. So it starts off as:
0, 1, 1, 2, 3, 5, 8, 13, .
The formula is:
F(0) = 0
F(1) = 1
F(n) = F(n-1) + F(n-2)
Step-by-Step Example (C Example: n = 5)
We need the first 5 terms:
F(0) = 0 (base case)
F(1) = 1 (base case)
F(2) = F(1) + F(0) = 1 + 0 = 1
F(3) = F(2) + F(1) = 1 + 1 = 2
F(4) = F(3) + F(2) = 2 + 1 = 3
The result is: 0 1 1 2 3
Real-Life Analogy
Picture a rabbit family tree. You begin with an infant rabbit in month 0. At month 1, it matures. Starting at month 2, every rabbit gives birth to a new rabbit each month, and so on for every new one that has been born. The growth exactly follows the Fibonacci sequence — the population every month is the addition of the two months previously.
Why Recursion Here
The recursion recursively defines itself in terms of smaller sub-problems. To compute F(n), you need to know F(n-1) and F(n-2). The relationship comes straight out as recursion exactly matches the mathematical definition.
Complexity
The recursive solution given here takes O(2^n) time since it repeatedly calculates many values. For high n, this function would be very slow. In real life, iterative or memoization solutions are used for performance. Space is O(n) due to the depth of the recursion stack.
Common Pitfalls
Omitting base cases (n <= 1). The recursion never terminates without them.
Using recursion for large n without optimization — you’ll get a huge slowdown or even stack overflow.
Confusing “first n terms” with “nth term.”
Practical Uses
The Fibonacci sequence appears in computer algorithms, financial modeling, population growth studies, and even nature (spirals in shells, arrangement of leaves). In coding interviews, it’s a favorite for testing recursion, dynamic programming, and optimization skills.
Learning Insights
Writing Fibonacci recursively is something that can teach you recurrence relations, base case considerations, and the distinction between exponential time and linear time algorithms. It's also an entry point into dynamic programming once you see how it can be optimized.
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The recursive Fibonacci sequence is among the most traditional programming exercises for beginners and job interview practice. It imparts essential recursion, mathematical solution, and trade-off efficiency concepts. Mastering the recursive Fibonacci algorithm in C, C++, Java, and Python allows programmers to establish a foundation for more advanced subjects such as dynamic programming, divide-and-conquer algorithms, and numerical mathematics. Such knowledge is crucial for competitive programming and technical interview employment.
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