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Sum of square of first n even numbers

Last Updated : 22 Sep, 2022
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Given a number n, find sum of square of first n even natural numbers. 

Examples:

Input : 3
Output : 56 
22 + 42 + 62 = 56

Input : 8
Output : 816
22 + 42 + 62 + 82 + 102 + 122 + 142 + 162 = 816

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

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

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

// Driver Code
int main()
{
    cout << squareSum(8);
    return 0;
}
Java 
// Simple Java method to find sum of 
// square of first n even numbers.
import java.io.*;

class GFG 
{
    static int squareSum(int n)
    {
        int sum = 0;
        for (int i = 1; i <= n; i++)
            sum += (2 * i) * (2 * i);
        return sum;
    }
    
    // Driver Code
    public static void main(String args[])
                  throws IOException
    {
        System.out.println(squareSum(8));
    }
}

// This code is contributed by Nikita Tiwari
Python3 
# Simple Python3 code to 
# find sum of square of
# first n even numbers

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

# driver code
ans = squareSum(8)
print (ans)

# This code is contributed by Saloni Gupta
C# 
// Simple C# method to find sum of
// square of first n even numbers.
using System;

class GFG 
{
    static int squareSum(int n)
    {
        int sum = 0;
        for (int i = 1; i <= n; i++)
            sum += (2 * i) * (2 * i);
        
        return sum;
    }
    
    // Driver code
    public static void Main()
    {
        Console.Write(squareSum(8));
    }
}

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

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


    echo squareSum(8);


// This code is contributed by vt_m.
?>
JavaScript 
<script>

// Simple Javascript method to find sum 
// of square of first n even numbers.
function squareSum(n)
{
    sum = 0;
    for(let i = 1; i <= n; i++)
        sum += (2 * i) * (2 * i);
        
    return sum;
}

// Driver Code
document.write(squareSum(8));

// This code is contributed by souravmahato348

</script>

Output: 

816

Time Complexity: O(n), where n represents the given integer.
Auxiliary Space: O(1), no extra space is required, so it is a constant.

An efficient solution is to apply below formula.

sum = 2 * n * (n + 1) * (2 * n + 1)/3

How does it work? 
We know that sum of square of first 
n natural numbers is = n(n+1)/2

Sum of square of first n even natural numbers = 
        22 + 42 + .... + (2n)2
      = 4 * (12 + 22 + .... + n2)
      = 4 * n(n+1)(2n+1) / 6
      = 2 * n(n+1)(2n+1)/3
C++ 
// Efficient C++ method to find sum 
// of square of first n even numbers.
#include <iostream>
using namespace std;

int squareSum(int n)
{
    return 2 * n * (n + 1) * 
            (2 * n + 1) / 3;
}

// Driver code
int main()
{
    cout << squareSum(8);
    return 0;
}
Java 
// Efficient Java method to find sum 
// of square of first n even numbers.
import java.io.*;

class GFG 
{
    static int squareSum(int n)
    {
        return 2 * n * (n + 1) * 
                (2 * n + 1) / 3;
    }

    // Driver Code
    public static void main(String args[])
                    throws IOException
    {
        System.out.println(squareSum(8));
    }
}

// This code is contributed by Nikita Tiwari
Python3 
# Efficient Python3 code to 
# find sum of square of 
# first n even numbers

def squareSum( n ):
    
    return int(2 * n * (n + 1) * (2 * n + 1) / 3)

# driver code
ans = squareSum(8)
print (ans)

# This code is contributed by Saloni Gupta
C# 
// Efficient C# method to find sum 
// of square of first n even numbers.
using System;

class GFG
{
    
    static int squareSum(int n)
    {
        return 2 * n * (n + 1) * (2 * n + 1) / 3;
    }
    
    // driver code
    public static void Main()
    {
        Console.Write(squareSum(8));
    }
}

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


function squareSum( $n)
{
    return 2 * $n * ($n + 1) * 
             (2 * $n + 1) / 3;
}

echo squareSum(8);

// This code is contributed by vt_m.
?>
JavaScript 
<script>

// JavaScript program to find sum of
// square of first n even numbers.

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

// Driver code

            document.write(squareSum(8));

</script>

Output : 

816

Time complexity: O(1) since performing constant operations
Auxiliary Space: O(1) because constant extra space has been used


D

dharmendra_kumar
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