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Code Mathematics

Sum of squares of first n natural numbers

The problem at hand involves calculating the sum of the squares of the first n natural numbers. Natural numbers are the positive integers starting from 1. The task is to find the sum of the squares of these natural numbers.

Problem Statement

Given a positive integer n, the objective is to calculate the sum of the squares of the first n natural numbers.

Example

For n = 4, the first n natural numbers are 1, 2, 3, and 4. The squares of these numbers are 1, 4, 9, and 16. The sum of these squares is 1 + 4 + 9 + 16 = 30.

Idea to Solve

To solve this problem, you can either directly calculate the sum using a loop or use a mathematical formula for the sum of squares. The sum of squares of the first n natural numbers can be calculated using the formula: sum = n * (n + 1) * (2n + 1) / 6.

Pseudocode

function squaresOfNaturalNo(n)
    sum = n * (n + 1) * (2 * n + 1) / 6
    print "Sum of square in", n, "natural numbers is", sum

Algorithm Explanation

  1. Calculate the sum of the squares of the first n natural numbers using the formula sum = n * (n + 1) * (2 * n + 1) / 6.
  2. Print the result.

Code Solution

/*
    C program for
    Sum of squares of first n natural numbers
*/
#include <stdio.h>

void squaresOfNaturalNo(int n)
{
	long long int sum = 0;
	if (n > 0)
	{
		// Normal calculation
		// Execute loop 1 to n and calculate squared.
		for (int i = 1; i <= n; ++i)
		{
			// Add square
			sum += (i *i);
		}
	}
	printf("\n Sum of square in %d natural number is %lld", n, sum);
}
int main(int argc, char const *argv[])
{
	/*
	   Test A
	   n = 5
	   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	*/
	squaresOfNaturalNo(5);
	/*
	   Test B
	   n = 8
	   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	   (7)² + (8)²  = 204
	*/
	squaresOfNaturalNo(8);
	return 0;
}

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
/*
    Java program for
    Sum of squares of first n natural numbers
*/
public class NaturalNumbers
{
	public void squaresOfNaturalNo(int n)
	{
		long sum = 0;
		if (n > 0)
		{
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			for (int i = 1; i <= n; ++i)
			{
				// Add square
				sum += (i * i);
			}
		}
		System.out.print("\n Sum of square in " + 
                         n + " natural number is " + sum);
	}
	public static void main(String[] args)
	{
		NaturalNumbers task = new NaturalNumbers();
		/*
		   Test A
		   n = 5
		   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
		    
		*/
		task.squaresOfNaturalNo(5);
		/*
		   Test B
		   n = 8
		   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
		   (7)² + (8)²  = 204
		*/
		task.squaresOfNaturalNo(8);
	}
}

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
// Include header file
#include <iostream>

using namespace std;
/*
    C++ program for
    Sum of squares of first n natural numbers
*/
class NaturalNumbers
{
	public: void squaresOfNaturalNo(int n)
	{
		long sum = 0;
		if (n > 0)
		{
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			for (int i = 1; i <= n; ++i)
			{
				// Add square
				sum += (i *i);
			}
		}
		cout << "\n Sum of square in " << n 
      		 << " natural number is " << sum;
	}
};
int main()
{
	NaturalNumbers *task = new NaturalNumbers();
	/*
	   Test A
	   n = 5
	   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	    
	*/
	task->squaresOfNaturalNo(5);
	/*
	   Test B
	   n = 8
	   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	   (7)² + (8)²  = 204
	*/
	task->squaresOfNaturalNo(8);
	return 0;
}

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
// Include namespace system
using System;
/*
    Csharp program for
    Sum of squares of first n natural numbers
*/
public class NaturalNumbers
{
	public void squaresOfNaturalNo(int n)
	{
		long sum = 0;
		if (n > 0)
		{
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			for (int i = 1; i <= n; ++i)
			{
				// Add square
				sum += (i * i);
			}
		}
		Console.Write("\n Sum of square in " + n + 
                      " natural number is " + sum);
	}
	public static void Main(String[] args)
	{
		NaturalNumbers task = new NaturalNumbers();
		/*
		   Test A
		   n = 5
		   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
		    
		*/
		task.squaresOfNaturalNo(5);
		/*
		   Test B
		   n = 8
		   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
		   (7)² + (8)²  = 204
		*/
		task.squaresOfNaturalNo(8);
	}
}

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
package main
import "fmt"
/*
    Go program for
    Sum of squares of first n natural numbers
*/

func squaresOfNaturalNo(n int) {
	var sum int64 = 0
	if n > 0 {
		// Normal calculation
		// Execute loop 1 to n and calculate squared.
		for i := 1 ; i <= n ; i++ {
			// Add square
			sum += int64(i * i)
		}
	}
	fmt.Print("\n Sum of square in ", n,
			 " natural number is ", sum)
}
func main() {

	/*
	   Test A
	   n = 5
	   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	    
	*/
	squaresOfNaturalNo(5)
	/*
	   Test B
	   n = 8
	   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	   (7)² + (8)²  = 204
	*/
	squaresOfNaturalNo(8)
}

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
<?php
/*
    Php program for
    Sum of squares of first n natural numbers
*/
class NaturalNumbers
{
	public	function squaresOfNaturalNo($n)
	{
		$sum = 0;
		if ($n > 0)
		{
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			for ($i = 1; $i <= $n; ++$i)
			{
				// Add square
				$sum += ($i * $i);
			}
		}
		echo("\n Sum of square in ".$n.
			" natural number is ".$sum);
	}
}

function main()
{
	$task = new NaturalNumbers();
	/*
	   Test A
	   n = 5
	   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	    
	*/
	$task->squaresOfNaturalNo(5);
	/*
	   Test B
	   n = 8
	   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	   (7)² + (8)²  = 204
	*/
	$task->squaresOfNaturalNo(8);
}
main();

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
/*
    Node JS program for
    Sum of squares of first n natural numbers
*/
class NaturalNumbers
{
	squaresOfNaturalNo(n)
	{
		var sum = 0;
		if (n > 0)
		{
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			for (var i = 1; i <= n; ++i)
			{
				// Add square
				sum += (i * i);
			}
		}
		process.stdout.write("\n Sum of square in " + n + 
                             " natural number is " + sum);
	}
}

function main()
{
	var task = new NaturalNumbers();
	/*
	   Test A
	   n = 5
	   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	    
	*/
	task.squaresOfNaturalNo(5);
	/*
	   Test B
	   n = 8
	   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	   (7)² + (8)²  = 204
	*/
	task.squaresOfNaturalNo(8);
}
main();

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
#    Python 3 program for
#    Sum of squares of first n natural numbers
class NaturalNumbers :
	def squaresOfNaturalNo(self, n) :
		sum = 0
		if (n > 0) :
			i = 1
			#  Normal calculation
			#  Execute loop 1 to n and calculate squared.
			while (i <= n) :
				#  Add square
				sum += (i * i)
				i += 1
			
		
		print("\n Sum of square in", 
              n ,"natural number is ", 
              sum, end = "")
	

def main() :
	task = NaturalNumbers()
	#   Test A
	#   n = 5
	#   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	task.squaresOfNaturalNo(5)
	#   Test B
	#   n = 8
	#   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	#   (7)² + (8)²  = 204
	task.squaresOfNaturalNo(8)

if __name__ == "__main__": main()

Output

 Sum of square in 5 natural number is  55
 Sum of square in 8 natural number is  204
#    Ruby program for
#    Sum of squares of first n natural numbers
class NaturalNumbers 
	def squaresOfNaturalNo(n) 
		sum = 0
		if (n > 0) 
			i = 1
			#  Normal calculation
			#  Execute loop 1 to n and calculate squared.
			while (i <= n) 
				#  Add square
				sum += (i * i)
				i += 1
			end

		end

		print("\n Sum of square in ", 
              n ," natural number is ", sum)
	end

end

def main() 
	task = NaturalNumbers.new()
	#   Test A
	#   n = 5
	#   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	task.squaresOfNaturalNo(5)
	#   Test B
	#   n = 8
	#   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	#   (7)² + (8)²  = 204
	task.squaresOfNaturalNo(8)
end

main()

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
/*
    Scala program for
    Sum of squares of first n natural numbers
*/
class NaturalNumbers()
{
	def squaresOfNaturalNo(n: Int): Unit = {
		var sum: Long = 0;
		if (n > 0)
		{
			var i: Int = 1;
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			while (i <= n)
			{
				// Add square
				sum += (i * i);
				i += 1;
			}
		}
		print("\n Sum of square in " + n + 
              " natural number is " + sum);
	}
}
object Main
{
	def main(args: Array[String]): Unit = {
		var task: NaturalNumbers = new NaturalNumbers();
		/*
		   Test A
		   n = 5
		   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
		    
		*/
		task.squaresOfNaturalNo(5);
		/*
		   Test B
		   n = 8
		   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
		   (7)² + (8)²  = 204
		*/
		task.squaresOfNaturalNo(8);
	}
}

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204
/*
    Swift 4 program for
    Sum of squares of first n natural numbers
*/
class NaturalNumbers
{
	func squaresOfNaturalNo(_ n: Int)
	{
		var sum: Int = 0;
		if (n > 0)
		{
			var i: Int = 1;
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			while (i <= n)
			{
				// Add square
				sum += (i * i);
				i += 1;
			}
		}
		print("\n Sum of square in", n ,
              "natural number is ", sum, terminator: "");
	}
}
func main()
{
	let task: NaturalNumbers = NaturalNumbers();
	/*
	   Test A
	   n = 5
	   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	    
	*/
	task.squaresOfNaturalNo(5);
	/*
	   Test B
	   n = 8
	   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	   (7)² + (8)²  = 204
	*/
	task.squaresOfNaturalNo(8);
}
main();

Output

 Sum of square in 5 natural number is  55
 Sum of square in 8 natural number is  204
/*
    Kotlin program for
    Sum of squares of first n natural numbers
*/
class NaturalNumbers
{
	fun squaresOfNaturalNo(n: Int): Unit
	{
		var sum: Long = 0;
		if (n > 0)
		{
			var i: Int = 1;
			// Normal calculation
			// Execute loop 1 to n and calculate squared.
			while (i <= n)
			{
				// Add square
				sum += (i * i);
				i += 1;
			}
		}
		print("\n Sum of square in " + n + 
              " natural number is " + sum);
	}
}
fun main(args: Array < String > ): Unit
{
	val task: NaturalNumbers = NaturalNumbers();
	/*
	   Test A
	   n = 5
	   (1)² + (2)² + (3)² + (4)² + (5)²  = 55
	    
	*/
	task.squaresOfNaturalNo(5);
	/*
	   Test B
	   n = 8
	   (1)² + (2)² + (3)² + (4)² + (5)² + (6)² + 
	   (7)² + (8)²  = 204
	*/
	task.squaresOfNaturalNo(8);
}

Output

 Sum of square in 5 natural number is 55
 Sum of square in 8 natural number is 204

Time Complexity

The time complexity of this algorithm is O(n), as it involves a loop that iterates from 1 to n to calculate the sum. The space complexity is constant, as it uses a fixed number of variables regardless of the input size.

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