Sum of Fibonacci Numbers

Last Updated : 13 Feb, 2023

Given a number positive number n, find value of f₀ + f₁ + f₂ + …. + f_n where f_i indicates i’th Fibonacci number. Remember that f₀ = 0, f₁ = 1, f₂ = 1, f₃ = 2, f₄ = 3, f₅ = 5, …
Examples :

Input : n = 3
Output : 4
Explanation : 0 + 1 + 1 + 2 = 4

Input : n = 4
Output : 7
Explanation : 0 + 1 + 1 + 2 + 3 = 7

Method 1 (O(n))
Brute Force approach is pretty straightforward, find all the Fibonacci numbers till f(n) and then add them up.

C++

// C++ Program to find sum of Fibonacci numbers
#include<bits/stdc++.h>
using namespace std;
 
// Computes value of first fibonacci numbers
int calculateSum(int n)
{
    if (n <= 0)
       return 0;
 
    int fibo[n+1];
    fibo[0] = 0, fibo[1] = 1;
 
    // Initialize result
    int sum = fibo[0] + fibo[1];
 
    // Add remaining terms
    for (int i=2; i<=n; i++)
    {
        fibo[i] = fibo[i-1]+fibo[i-2];
        sum += fibo[i];
    }
 
    return sum;
}
 
// Driver program to test above function
int main()
{
    int n = 4;
    cout << "Sum of Fibonacci numbers is : "
         << calculateSum(n) << endl;
    return 0;
}

Java

// Java Program to find
// sum of Fibonacci numbers
 
import java.io.*;
 
class GFG {
     
    // Computes value of first
    // fibonacci numbers
    static int calculateSum(int n)
    {
        if (n <= 0)
           return 0;
      
        int fibo[]=new int[n+1];
        fibo[0] = 0; fibo[1] = 1;
      
        // Initialize result
        int sum = fibo[0] + fibo[1];
      
        // Add remaining terms
        for (int i=2; i<=n; i++)
        {
            fibo[i] = fibo[i-1]+fibo[i-2];
            sum += fibo[i];
        }
      
        return sum;
    }
      
    // Driver program to test above function
    public static void main(String args[])
    {
        int n = 4;
        System.out.println("Sum of Fibonacci" + 
        " numbers is : "+ calculateSum(n));
    }
}
 
// This code is contributed by Nikita tiwari.

Python3

# Python 3 Program to find
# sum of Fibonacci numbers
 
 
# Computes value of first
# fibonacci numbers
def calculateSum(n) :
    if (n <= 0) :
        return 0
  
    fibo =[0] * (n+1)
    fibo[1] = 1
  
    # Initialize result
    sm = fibo[0] + fibo[1]
  
    # Add remaining terms
    for i in range(2,n+1) :
        fibo[i] = fibo[i-1] + fibo[i-2]
        sm = sm + fibo[i]
         
    return sm
 
 
# Driver program to test
# above function
n = 4
print("Sum of Fibonacci numbers is : " ,
      calculateSum(n))
 
# This code is contributed
# by Nikita tiwari.

C#

// C# Program to find
// sum of Fibonacci numbers
using System;
 
class GFG 
{
     
    // Computes value of first
    // fibonacci numbers
    static int calculateSum(int n)
    {
        if (n <= 0)
        return 0;
     
        int []fibo = new int[n + 1];
        fibo[0] = 0; fibo[1] = 1;
     
        // Initialize result
        int sum = fibo[0] + fibo[1];
     
        // Add remaining terms
        for (int i = 2; i <= n; i++)
        {
            fibo[i] = fibo[i - 1] + fibo[i - 2];
            sum += fibo[i];
        }
     
        return sum;
    }
     
    // Driver Code
    static void Main()
    {
        int n = 4;
        Console.WriteLine( "Sum of Fibonacci" + 
                              " numbers is : "+ 
                              calculateSum(n));
    }
}
 
// This code is contributed by Anuj_67

PHP

<?php
// PHP Program to find sum 
// of Fibonacci numbers
 
// Computes value of first 
// fibonacci numbers
function calculateSum($n)
{
    if ($n <= 0)
    return 0;
 
    $fibo[0] = 0;
    $fibo[1] = 1;
 
    // Initialize result
    $sum = $fibo[0] + $fibo[1];
 
    // Add remaining terms
    for($i = 2; $i <= $n; $i++)
    {
        $fibo[$i] = $fibo[$i - 1] + 
                    $fibo[$i - 2];
        $sum += $fibo[$i];
    }
 
    return $sum;
}
 
    // Driver Code
    $n = 4;
    echo "Sum of Fibonacci numbers is : ",
          calculateSum($n),"\n";
 
// This code is contributed by aj_36
?>

Javascript

<script>
// Javascript Program to find sum
// of Fibonacci numbers
 
// Computes value of first
// fibonacci numbers
function calculateSum(n)
{
    let fibo = [];
    if (n <= 0)
    return 0;
 
    fibo[0] = 0;
    fibo[1] = 1;
 
    // Initialize result
    let sum = fibo[0] + fibo[1];
 
    // Add remaining terms
    for(let i = 2; i <= n; i++)
    {
        fibo[i] = fibo[i - 1] +
                    fibo[i - 2];
        sum += fibo[i];
    }
 
    return sum;
}
 
    // Driver Code
    let n = 4;
    document.write(`Sum of Fibonacci numbers is : 
        ${calculateSum(n)} <br>`);
 
// This code is contributed by _saurabh_jaiswal
</script>

Output

Sum of Fibonacci numbers is : 7

Time Complexity: O(n)
Auxiliary Space: O(n)

Method 2 : Without using extra space

C++

#include <iostream>
using namespace std;
 
int main() {
 
    int n = 4, a = 0, b = 0, sumf = 1; 
    if (n<=0)
      sumf = 0;
    int curr = 1;
    for(int i = 1;i<n;i++){
        a = b;
        b = curr;
        curr = a+b;
        sumf += curr;
    }
cout<<"The sum of fibonocci numbers is:"<<sumf<<endl;
    return 0;
}

Java

import java.io.*;
 
class GFG 
{
    public static void main(String args[])
    {
        int n = 4, a = 0, b = 0, sumf = 1; 
        if (n <= 0)
          sumf = 0;
        int curr = 1;
        for(int i = 1; i < n; i++){
            a = b;
            b = curr;
            curr = a+b;
            sumf += curr;
        }
    System.out.println("The sum of fibonocci numbers is:"+sumf);
    }
}

Python3

#By using this code we can calculate sum of 
#fibonocci numbers without using extra space
n = 4
a = 0
b = 0
sumf = 1
if n<=0:
  sumf = 0
curr = 1
for i in range(1,n):
  a = b
  b = curr
  curr = a+b
  sumf += curr
print("The sum of fibonocci numbers is:",sumf)

Javascript

let n = 4, a = 0, b = 0, sumf = 1; 
if (n<=0)
  sumf = 0;
let curr = 1;
for(let i = 1;i<n;i++){
    a = b;
    b = curr;
    curr = a+b;
    sumf += curr;
}
console.log("The sum of fibonocci numbers is:"+sumf);
  

C#

using System;
using System.Collections.Generic;
 
class Gfg
{
    public static void Main(string[] args)
    {
        int n = 4, a = 0, b = 0, sumf = 1; 
        if (n<=0)
            sumf = 0;
        int curr = 1;
        for(int i = 1;i<n;i++){
            a = b;
            b = curr;
            curr = a+b;
            sumf += curr;
        }
        Console.Write("The sum of fibonocci numbers is:"+sumf);
    }
}

Output

The sum of fibonocci numbers is:7

Time Complexity: O(n)
Auxiliary Space: O(1)

Method 3 (O(Log n))
The idea is to find relationship between the sum of Fibonacci numbers and n’th Fibonacci number.
F(i) refers to the i’th Fibonacci number.
S(i) refers to sum of Fibonacci numbers till F(i),

We can rewrite the relation F(n+1) = F(n) + F(n-1) as below
F(n-1)    = F(n+1)  -  F(n)

Similarly,
F(n-2)    = F(n)    -  F(n-1)
.          .           .
.          .             .
.          .             .
F(0)      = F(2)    -  F(1)
-------------------------------

Adding all the equations, on left side, we have
F(0) + F(1) + … F(n-1) which is S(n-1).
Therefore,
S(n-1) = F(n+1) – F(1)
S(n-1) = F(n+1) – 1
S(n) = F(n+2) – 1 —-(1)
In order to find S(n), simply calculate the (n+2)’th Fibonacci number and subtract 1 from the result.
F(n) can be evaluated in O(log n) time using either method 5 or method 6 in this article (Refer to methods 5 and 6).
Below is the implementation based on method 6 of this

C++

// C++ Program to find sum of Fibonacci numbers in 
// O(Log n) time.
#include <bits/stdc++.h>
using namespace std;
const int MAX = 1000;
 
// Create an array for memoization
int f[MAX] = {0};
 
// Returns n'th Fibonacci number using table f[]
int fib(int n)
{
    // Base cases
    if (n == 0)
        return 0;
    if (n == 1 || n == 2)
        return (f[n] = 1);
 
    // If fib(n) is already computed
    if (f[n])
        return f[n];
 
    int k = (n & 1)? (n+1)/2 : n/2;
 
    // Applying above formula [Note value n&1 is 1
    // if n is odd, else 0].
    f[n] = (n & 1)? (fib(k)*fib(k) + fib(k-1)*fib(k-1))
           : (2*fib(k-1) + fib(k))*fib(k);
 
    return f[n];
}
 
// Computes value of first Fibonacci numbers
int calculateSum(int n)
{
    return fib(n+2) - 1;
}
 
// Driver program to test above function
int main()
{
    int n = 4;
    cout << "Sum of Fibonacci numbers is : "
         << calculateSum(n) << endl;
    return 0;
}

Java

// Java Program to find sum of Fibonacci numbers in 
// O(Log n) time.
import java.util.*;
 
class GFG{
static int MAX = 1000;
 
// Create an array for memoization
static int []f = new int[MAX];
 
// Returns n'th Fibonacci number using table f[]
static int fib(int n)
{
   
    // Base cases
    if (n == 0)
    return 0;
    if (n == 1 || n == 2)
        return (f[n] = 1);
 
    // If fib(n) is already computed
    if (f[n]>0)
        return f[n];
 
    int k = ((n & 1)>0)? (n+1)/2 : n/2;
 
    // Applying above formula [Note value n&1 is 1
    // if n is odd, else 0].
    f[n] = (n & 1)>0? (fib(k)*fib(k) + fib(k-1)*fib(k-1))
           : (2*fib(k-1) + fib(k))*fib(k);
 
    return f[n];
}
 
// Computes value of first Fibonacci numbers
static int calculateSum(int n)
{
    return fib(n+2) - 1;
}
 
// Driver program to test above function
public static void main(String[] args)
{
    int n = 4;
    System.out.print("Sum of Fibonacci numbers is : "
         + calculateSum(n) +"\n");
}
}
 
// This code is contributed by gauravrajput1

Python3

# Python 3 Program to find sum of 
# Fibonacci numbers in O(Log n) time.
 
MAX = 1000
 
# Create an array for memoization
f = [0] * MAX
 
# Returns n'th Fibonacci number
# using table f[]
def fib(n):
     
    n = int(n)
 
    # Base cases
    if (n == 0):
        return 0
    if (n == 1 or n == 2):
        return (1) 
 
    # If fib(n) is already computed
    if (f[n] == True):
        return f[n] 
 
    k = (n+1)/2 if (n & 1) else n/2
 
    # Applying above formula [Note value n&1 
    # is 1 if n is odd, else 0].
    f[n] = (fib(k) * fib(k) + fib(k-1) * fib(k-1)) if (n & 1) else (2 * fib(k-1) + fib(k)) * fib(k)
    return f[n] 
 
# Computes value of first Fibonacci numbers
def calculateSum(n):
 
    return fib(n+2) - 1
 
# Driver program to test above function
n = 4
print("Sum of Fibonacci numbers is :", calculateSum(n)) 
 
# This code is contributed by
# Smitha Dinesh Semwal

C#

// C# Program to find sum
// of Fibonacci numbers in 
// O(Log n) time.
using System;
 
class GFG {
    static int MAX = 1000;
 
    // Create an array for memoization
    static int []f = new int[MAX];
     
    // Returns n'th Fibonacci 
    // number using table f[]
    static int fib(int n)
    {
        for(int i = 0;i < MAX;i++)
        f[i] = 0;
         
        //Arrays.fill(f, 0);
        // Base cases
        if (n == 0)
            return 0;
        if (n == 1 || n == 2)
            return (f[n] = 1);
     
        // If fib(n) is 
        // already computed
        if (f[n] == 1)
            return f[n];
            int k;
        if((n & 1) == 1) 
            k = (n + 1) / 2 ;
        else
            k = n / 2;
     
        // Applying above formula 
        // [Note value n&1 is 1 
        // if n is odd, else 0].
        if((n & 1) == 1) 
            f[n] = (fib(k) * fib(k) + fib(k - 1)
                                   * fib(k - 1));
        else
            f[n] = (2 * fib(k - 1) + fib(k)) *
                                       fib(k);
     
        return f[n];
    }
     
    // Computes value of first 
    // Fibonacci numbers
    static int calculateSum(int n)
    {
        return fib(n + 2) - 1;
    }
     
    // Driver Code
    public static void Main()
    {
        int n = 4;
        Console.Write( "Sum of Fibonacci numbers is : "
                                    + calculateSum(n));
         
    }
}
 
// This code is contributed by nitin mittal.

PHP

<?php
// PHP Program to find sum of Fibonacci 
// numbers in O(Log n) time.
$MAX = 1000;
 
// Create an array for memoization
$f = array_fill(0, $MAX, 0);
 
// Returns n'th Fibonacci number 
// using table f[]
function fib($n)
{
    global $f;
     
    // Base cases
    if ($n == 0)
        return 0;
    if ($n == 1 || $n == 2)
        return ($f[$n] = 1);
 
    // If fib(n) is already computed
    if ($f[$n])
        return $f[$n];
 
    $k = ($n & 1) ? ($n + 1) / 2 : $n / 2;
 
    // Applying above formula [Note value n&1 
    // is 1 if n is odd, else 0].
    $f[$n] = ($n & 1) ? 
             (fib($k) * fib($k) + fib($k - 1) * fib($k - 1)) : 
                  (2 * fib($k - 1) + fib($k)) * fib($k);
 
    return $f[$n];
}
 
// Computes value of first Fibonacci numbers
function calculateSum($n)
{
    return fib($n + 2) - 1;
}
 
// Driver Code
$n = 4;
print("Sum of Fibonacci numbers is : " . 
                      calculateSum($n));
 
// This code is contributed by mits
?>

Javascript

<script>
// javascript Program to find sum of Fibonacci numbers in 
// O(Log n) time.
    var MAX = 1000;
 
    // Create an array for memoization
     var f = Array(MAX).fill(0);
 
    // Returns n'th Fibonacci number using table f
    function fib(n) {
 
        // Base cases
        if (n == 0)
            return 0;
        if (n == 1 || n == 2)
            return (f[n] = 1);
 
        // If fib(n) is already computed
        if (f[n] > 0)
            return f[n];
 
        var k = ((n & 1) > 0) ? (n + 1) / 2 : n / 2;
 
        // Applying above formula [Note value n&1 is 1
        // if n is odd, else 0].
        f[n] = (n & 1) > 0 ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1)) : (2 * fib(k - 1) + fib(k)) * fib(k);
 
        return f[n];
    }
 
    // Computes value of first Fibonacci numbers
    function calculateSum(n) {
        return fib(n + 2) - 1;
    }
 
    // Driver program to test above function
        var n = 4;
        document.write("Sum of Fibonacci numbers is : " + calculateSum(n) + "\n");
 
// This code is contributed by gauravrajput1
</script>

Output

Sum of Fibonacci numbers is : 7

Time Complexity: O(logn)
Auxiliary Space: O(MAX)

Suggest improvement

Tail Recursion for Fibonacci

Fibonacci Series

Share your thoughts in the comments

Fibonacci Series

Variations of Fibonacci number

Some other problems on Fibonacci Numbers

Sum of Fibonacci Numbers

C++

Java

Python3

C#

PHP

Javascript

C++

Java

Python3

Javascript

C#

C++

Java

Python3

C#

PHP

Javascript

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