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Sum of cubes of first n even numbers

Last Updated : 16 Feb, 2023
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Given a number n, find the sum of first n even natural numbers. 

Examples: 

Input : 2
Output : 72
2^3 + 4^3 = 72
Input : 8
Output :10368
2^3 + 4^3 + 6^3 + 8^3 + 10^3 + 12^3 + 14^3 + 16^3 = 10368

A simple solution is to traverse through n even numbers and find the sum of cubes.  

C++ 
// Simple C++ method to find sum of cubes of
// first n even numbers.
#include <iostream>
using namespace std;

int cubeSum(int n)
{
    int sum = 0;
    for (int i = 1; i <=  n; i++)
        sum += (2*i) * (2*i) * (2*i);
    return sum;
}

int main()
{
    cout << cubeSum(8);
    return 0;
}
Java 
// Java program to perform 
// sum of cubes of first 
// n even natural numbers

public class GFG 
{
    public static int cubesum(int n)
    {
        int sum = 0;
        for(int i = 1; i <= n; i++)
            sum += (2 * i) * (2 * i) 
                   * (2 * i);
                
        return sum;
    }
    

    // Driver function
    public static void main(String args[])
    {
        int a = 8;
        System.out.println(cubesum(a));
        
    }
}

// This code is contributed by Akansh Gupta
Python3 
# Python3 program to find sum of 
# cubes of first n even numbers

# Function to find sum of cubes
# of first n even numbers 
def cubeSum(n):

    sum = 0
    for i in range(1, n + 1):
        sum += (2 * i) * (2 * i) * (2 * i)
    return sum

# Driven code 
print(cubeSum(8))

# This code is contributed by Shariq Raza
C# 
// C# program to perform 
// sum of cubes of first 
// n even natural numbers
using System;

public class GFG 
{
    public static int cubesum(int n)
    {
        int sum = 0;
        for(int i = 1; i <= n; i++)
            sum += (2 * i) * (2 * i) 
                * (2 * i);
                
        return sum;
    }
    

    // Driver function
    public static void Main()
    {
        int a = 8;
        Console.WriteLine(cubesum(a));
        
    }
}

// This code is contributed by vt_m.
PHP 
<?php
// Simple PHP method to
// find sum of cubes of
// first n even numbers.

function cubeSum($n)
{
    $sum = 0;
    for ($i = 1; $i <= $n; $i++)
        $sum += (2 * $i) * 
                (2 * $i) * 
                (2 * $i);
    return $sum;
}

// Driver Code
echo cubeSum(8);
    
// This code is contributed by vt_m.
?>
JavaScript 
<script>
// JavaScript program to find sum of cubes of 
// first n even numbers. 

    function cubeSum(n) 
    { 
        let sum = 0; 
        for (let i = 1; i <=  n; i++) 
        sum += (2*i) * (2*i) * (2*i); 
        return sum; 
    } 
  
    document.write(cubeSum(8));  

// This code is contributed by Surbhi Tyagi

</script>

Output:

10368

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

An efficient solution is to apply below formula. 

sum = 2 *  n2(n+1)2 
 

How does it work? 

We know that sum of cubes of first 
n natural numbers is = n2(n+1)2 / 4

Sum of cubes of first n natural numbers = 
                2^3 + 4^3 + .... + (2n)^3
              = 8 * (1^3 + 2^3 + .... + n^3)
              = 8 *  n2(n+1)2 / 4
              = 2 *  n2(n+1)2

Example

C++ 
// Efficient C++ method to find sum of cubes of
// first n even numbers.
#include <iostream>
using namespace std;

int cubeSum(int n)
{
    return 2 * n * n * (n + 1) * (n + 1);
}

int main()
{
    cout << cubeSum(8);
    return 0;
}
Java 
// Java program to perform
// sum of cubes of first 
// n even natural numbers

public class GFG
{
    public static int cubesum(int n)
    {
                
        return 2 * n * n * (n + 1) * (n + 1);
    }
    

    // Driver function
    public static void main(String args[])
    {
        int a = 8;
        System.out.println(cubesum(a));
        
    }
}

// This code is contributed by Akansh Gupta
Python3 
# Python3 program to find sum of
# cubes of first n even numbers

# Function to find sum of cubes
# of first n even numbers 
def cubeSum(n):
    
    return 2 * n * n * (n + 1) * (n + 1)

# Driven code 
print(cubeSum(8))

# This code is contributed by Shariq Raza
C# 
// C# program to perform
// sum of cubes of first 
// n even natural numbers
using System;

class GFG
{
    public static int cubesum(int n)
    {
        return 2 * n * n * 
               (n + 1) * (n + 1);
    }
    
    // Driver code
    public static void Main()
    {
        int a = 8;
        Console.WriteLine(cubesum(a));
    }
}

// This code is contributed by vt_m.
PHP 
<?php
// Efficient PHP code to 
// find sum of cubes of
// first n even numbers.

function cubeSum($n)
{
    return 2 * $n * $n * 
           ($n + 1) * ($n + 1);
}

// Driver code
echo cubeSum(8);
    
// This code is contributed by vt_m.
?>
JavaScript 
<script>
// javascript program to perform
// sum of cubes of first 
// n even natural numbers
function cubesum(n)
{     
    return 2 * n * n * (n + 1) * (n + 1);
}

// Driver function
var a = 8;
document.write(cubesum(a));

// This code is contributed by Amit Katiyar 
</script>

Output:

10368

Time Complexity: O(1)

Auxiliary Space: O(1)

Sum of cube of first n odd natural numbers


D

Dharmendra kumar
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