Average of squares of natural numbers
The problem entails calculating the average of the squares of the first n natural numbers. Natural numbers are the positive integers starting from 1. The task is to find the average value of the squares of these natural numbers.
Problem Statement
Given a positive integer n, the goal is to calculate the average 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 average of these squares is (1 + 4 + 9 + 16) / 4 = 30 / 4 = 7.5.
Idea to Solve
To solve this problem, one can utilize the sum formula for the squares of consecutive natural numbers. The formula is
given by: 1^2 + 2^2 + 3^2 + ... + n^2 = (n * (n + 1) * (2 * n + 1)) / 6. Then, the average can be
calculated by dividing the sum of squares by n.
Pseudocode
function averageOfNaturalNo(n)
sum = (n * (n + 1) * (2 * n + 1)) / 6
average = sum / n
print "Average of squares of", n, "natural numbers is:", averageAlgorithm Explanation
- Calculate the sum of squares of the first n natural numbers using the formula
sum = (n * (n + 1) * (2 * n + 1)) / 6. - Calculate the average by dividing the sum by n:
average = sum / n. - Print the result.
Code Solution
/*
C program for
Average of squares of natural numbers
*/
#include <stdio.h>
// Average of series of squares of natural numbers
void averageOfNaturalNo(int n)
{
long double sum = 0.0;
double result = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = ((n *(n + 1) *(2 *n + 1)) / 6);
// Calculate average
result = sum / n;
}
// Display calculated result
printf(" Average of squares of %d natural number is : %lf \n",
n, result);
}
int main(int argc, char
const *argv[])
{
/*
Test A
n = 5
(1)² + (2)² + (3)² + (4)² + (5)² = 55
55
-- = 11
5
*/
averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
averageOfNaturalNo(8);
return 0;
}Output
Average of squares of 5 natural number is : 11.000000
Average of squares of 8 natural number is : 25.500000/*
Java program for
Average of squares of natural numbers
*/
public class NaturalNumbers
{
// Average of series of squares of natural numbers
public void averageOfNaturalNo(int n)
{
double sum = 0.0;
double result = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = ((n * (n + 1) * (2 * n + 1)) / 6);
// Calculate average
result = sum / n;
}
// Display calculated result
System.out.print(" Average of squares of " + n + " natural number is : " + result + " \n");
}
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
55
-- = 11
5
*/
task.averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
task.averageOfNaturalNo(8);
}
}Output
Average of squares of 5 natural number is : 11.0
Average of squares of 8 natural number is : 25.5// Include header file
#include <iostream>
using namespace std;
/*
C++ program for
Average of squares of natural numbers
*/
class NaturalNumbers
{
public:
// Average of series of squares of natural numbers
void averageOfNaturalNo(int n)
{
double sum = 0.0;
double result = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = ((n *(n + 1) *(2 *n + 1)) / 6);
// Calculate average
result = sum / n;
}
// Display calculated result
cout << " Average of squares of "
<< n << " natural number is : "
<< result << " \n";
}
};
int main()
{
NaturalNumbers *task = new NaturalNumbers();
/*
Test A
n = 5
(1)² + (2)² + (3)² + (4)² + (5)² = 55
55
-- = 11
5
*/
task->averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
task->averageOfNaturalNo(8);
return 0;
}Output
Average of squares of 5 natural number is : 11
Average of squares of 8 natural number is : 25.5// Include namespace system
using System;
/*
Csharp program for
Average of squares of natural numbers
*/
public class NaturalNumbers
{
// Average of series of squares of natural numbers
public void averageOfNaturalNo(int n)
{
double sum = 0.0;
double result = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = ((n * (n + 1) * (2 * n + 1)) / 6);
// Calculate average
result = sum / n;
}
// Display calculated result
Console.Write(" Average of squares of " +
n + " natural number is : " +
result + " \n");
}
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
55
-- = 11
5
*/
task.averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
task.averageOfNaturalNo(8);
}
}Output
Average of squares of 5 natural number is : 11
Average of squares of 8 natural number is : 25.5package main
import "fmt"
/*
Go program for
Average of squares of natural numbers
*/
// Average of series of squares of natural numbers
func averageOfNaturalNo(n int) {
var sum float64 = 0.0
var result float64 = 0.0
if n > 0 {
// Calculate sum of squares of n natural numbers
sum = float64((n * (n + 1) * (2 * n + 1)) / 6.0)
// Calculate average
result = sum / float64(n)
}
// Display calculated result
fmt.Print(" Average of squares of ",
n, " natural number is : ", result, " \n")
}
func main() {
/*
Test A
n = 5
(1)² + (2)² + (3)² + (4)² + (5)² = 55
55
-- = 11
5
*/
averageOfNaturalNo(5)
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
averageOfNaturalNo(8)
}Output
Average of squares of 5 natural number is : 11.0
Average of squares of 8 natural number is : 25.5<?php
/*
Php program for
Average of squares of natural numbers
*/
class NaturalNumbers
{
// Average of series of squares of natural numbers
public function averageOfNaturalNo($n)
{
$sum = 0.0;
$result = 0.0;
if ($n > 0)
{
// Calculate sum of squares of n natural numbers
$sum = ((($n * ($n + 1) * (2 * $n + 1)) / 6));
// Calculate average
$result = $sum / $n;
}
// Display calculated result
echo(" Average of squares of ".$n.
" natural number is : ".$result.
" \n");
}
}
function main()
{
$task = new NaturalNumbers();
/*
Test A
n = 5
(1)² + (2)² + (3)² + (4)² + (5)² = 55
55
-- = 11
5
*/
$task->averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
$task->averageOfNaturalNo(8);
}
main();Output
Average of squares of 5 natural number is : 11
Average of squares of 8 natural number is : 25.5/*
Node JS program for
Average of squares of natural numbers
*/
class NaturalNumbers
{
// Average of series of squares of natural numbers
averageOfNaturalNo(n)
{
var sum = 0.0;
var result = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = (((n * (n + 1) * (2 * n + 1)) / 6));
// Calculate average
result = sum / n;
}
// Display calculated result
process.stdout.write(" Average of squares of " +
n + " natural number is : " +
result + " \n");
}
}
function main()
{
var task = new NaturalNumbers();
/*
Test A
n = 5
(1)² + (2)² + (3)² + (4)² + (5)² = 55
55
-- = 11
5
*/
task.averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
task.averageOfNaturalNo(8);
}
main();Output
Average of squares of 5 natural number is : 11
Average of squares of 8 natural number is : 25.5# Python 3 program for
# Average of squares of natural numbers
class NaturalNumbers :
# Average of series of squares of natural numbers
def averageOfNaturalNo(self, n) :
sum = 0.0
result = 0.0
if (n > 0) :
# Calculate sum of squares of n natural numbers
sum = (((n * (n + 1) * (2 * n + 1)) / 6))
# Calculate average
result = sum / n
# Display calculated result
print(" Average of squares of ",
n ," natural number is : ",
result ," ")
def main() :
task = NaturalNumbers()
# Test A
# n = 5
# (1)² + (2)² + (3)² + (4)² + (5)² = 55
# 55
# -- = 11
# 5
task.averageOfNaturalNo(5)
# Test B
# n = 8
# (1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
# (7)² + (8)² = 204
# 204
# ---- = 25.5
# 8
task.averageOfNaturalNo(8)
if __name__ == "__main__": main()Output
Average of squares of 5 natural number is : 11.0
Average of squares of 8 natural number is : 25.5# Ruby program for
# Average of squares of natural numbers
class NaturalNumbers
# Average of series of squares of natural numbers
def averageOfNaturalNo(n)
sum = 0.0
result = 0.0
if (n > 0)
# Calculate sum of squares of n natural numbers
sum = ((n * (n + 1) * (2 * n + 1)) / 6.0)
# Calculate average
result = sum / n
end
# Display calculated result
print(" Average of squares of ",
n ," natural number is : ", result ," \n")
end
end
def main()
task = NaturalNumbers.new()
# Test A
# n = 5
# (1)² + (2)² + (3)² + (4)² + (5)² = 55
# 55
# -- = 11
# 5
task.averageOfNaturalNo(5)
# Test B
# n = 8
# (1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
# (7)² + (8)² = 204
# 204
# ---- = 25.5
# 8
task.averageOfNaturalNo(8)
end
main()Output
Average of squares of 5 natural number is : 11.0
Average of squares of 8 natural number is : 25.5
/*
Scala program for
Average of squares of natural numbers
*/
class NaturalNumbers()
{
// Average of series of squares of natural numbers
def averageOfNaturalNo(n: Int): Unit = {
var sum: Double = 0.0;
var result: Double = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = ((n * (n + 1) * (2 * n + 1)) / 6);
// Calculate average
result = sum / n;
}
// Display calculated result
print(" Average of squares of " +
n + " natural number is : " + result + " \n");
}
}
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
55
-- = 11
5
*/
task.averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
task.averageOfNaturalNo(8);
}
}Output
Average of squares of 5 natural number is : 11.0
Average of squares of 8 natural number is : 25.5/*
Swift 4 program for
Average of squares of natural numbers
*/
class NaturalNumbers
{
// Average of series of squares of natural numbers
func averageOfNaturalNo(_ n: Int)
{
var sum: Double = 0.0;
var result: Double = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = Double((n * (n + 1) * (2 * n + 1)) / 6);
// Calculate average
result = sum / Double(n);
}
// Display calculated result
print(" Average of squares of",
n ,"natural number is : ", result ," ");
}
}
func main()
{
let task: NaturalNumbers = NaturalNumbers();
/*
Test A
n = 5
(1)² + (2)² + (3)² + (4)² + (5)² = 55
55
-- = 11
5
*/
task.averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
task.averageOfNaturalNo(8);
}
main();Output
Average of squares of 5 natural number is : 11.0
Average of squares of 8 natural number is : 25.5/*
Kotlin program for
Average of squares of natural numbers
*/
class NaturalNumbers
{
// Average of series of squares of natural numbers
fun averageOfNaturalNo(n: Int): Unit
{
var sum: Double ;
var result: Double = 0.0;
if (n > 0)
{
// Calculate sum of squares of n natural numbers
sum = ((n * (n + 1) * (2 * n + 1)) / 6.0);
// Calculate average
result = sum / n;
}
// Display calculated result
print(" Average of squares of " +
n + " natural number is : " +
result + " \n");
}
}
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
55
-- = 11
5
*/
task.averageOfNaturalNo(5);
/*
Test B
n = 8
(1)² + (2)² + (3)² + (4)² + (5)² + (6)² +
(7)² + (8)² = 204
204
---- = 25.5
8
*/
task.averageOfNaturalNo(8);
}Output
Average of squares of 5 natural number is : 11.0
Average of squares of 8 natural number is : 25.5Time Complexity
The time complexity of this algorithm is constant, O(1), as it involves only a few mathematical operations that do not depend on the input size. The space complexity is also constant, as it uses a fixed number of variables.
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