Calculate the sum of squares of the first N natural numbers in C#

In this problem, we are given a number N, and we need to calculate the sum of the squares of the first N natural numbers. The first N natural numbers are 1, 2, 3, ..., N, and we need to find 1² + 2² + 3² + ... + N².

Problem Description

Given a positive integer N, calculate the sum of squares of the first N natural numbers using different approaches in C#.

Example 1

• Input: N = 4
• Output: 30

Explanation:

The squares of the first 4 natural numbers are: 1², 2², 3², 4² = 1, 4, 9, 16.
The sum is: 1 + 4 + 9 + 16 = 30.

Example 2

• Input: N = 6
• Output: 91

Explanation:

The squares of the first 6 natural numbers are: 1, 4, 9, 16, 25, 36.
The sum is: 1 + 4 + 9 + 16 + 25 + 36 = 91.

Using an Iterative Approach

This is a straightforward approach where we iterate through the numbers from 1 to N, calculate their squares, and sum them up

using System;

class Program {
    static int SumOfSquaresIterative(int N) {
        int sum = 0;
        for (int i = 1; i <= N; i++) {
            sum += i * i; // Add square of the number to sum
        }
        return sum;
    }

    static void Main() {
        int N = 4;
        int result = SumOfSquaresIterative(N);
        Console.WriteLine("The sum of squares of the first {0} natural numbers is: {1}", N, result);
        
        // Test with another value
        N = 6;
        result = SumOfSquaresIterative(N);
        Console.WriteLine("The sum of squares of the first {0} natural numbers is: {1}", N, result);
    }
}

The output of the above code is

The sum of squares of the first 4 natural numbers is: 30
The sum of squares of the first 6 natural numbers is: 91

Time Complexity: O(N)
Space Complexity: O(1)

Using Mathematical Formula

This approach uses the mathematical formula to calculate the sum directly in O(1) time. The sum of squares of the first N natural numbers follows the formula

Sum = N × (N + 1) × (2N + 1) / 6

using System;

class Program {
    static int SumOfSquaresFormula(int N) {
        return (N * (N + 1) * (2 * N + 1)) / 6; // Mathematical formula
    }

    static void Main() {
        int N = 4;
        int result = SumOfSquaresFormula(N);
        Console.WriteLine("Using formula for N={0}: {1}", N, result);
        
        N = 6;
        result = SumOfSquaresFormula(N);
        Console.WriteLine("Using formula for N={0}: {1}", N, result);
        
        N = 10;
        result = SumOfSquaresFormula(N);
        Console.WriteLine("Using formula for N={0}: {1}", N, result);
    }
}

The output of the above code is

Using formula for N=4: 30
Using formula for N=6: 91
Using formula for N=10: 385

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

Using Recursive Approach

In this approach, we use recursion to calculate the sum of squares. The base case is when N = 0, and the recursive case adds the square of N to the sum of squares of the first N-1 numbers

using System;

class Program {
    static int SumOfSquaresRecursive(int N) {
        // Base case: if N is 0, sum is 0
        if (N == 0)
            return 0;

        // Recursive case: N² + sum of squares of (N-1)
        return (N * N) + SumOfSquaresRecursive(N - 1);
    }

    static void Main() {
        int N = 5;
        int result = SumOfSquaresRecursive(N);
        Console.WriteLine("Using recursion for N={0}: {1}", N, result);
        
        N = 3;
        result = SumOfSquaresRecursive(N);
        Console.WriteLine("Using recursion for N={0}: {1}", N, result);
    }
}

The output of the above code is

Using recursion for N=5: 55
Using recursion for N=3: 14

Time Complexity: O(N)
Space Complexity: O(N) due to recursive call stack

Comparison of Approaches

Common Use Cases

• Statistical Calculations: Used in computing variance and standard deviation in data analysis.
• Physics Simulations: Calculating kinetic energy, potential energy levels, and work done by forces.
• Machine Learning: Error calculations in regression analysis and optimization problems.
• Computer Graphics: Distance calculations and geometric transformations.

Conclusion

The sum of squares of first N natural numbers can be calculated using three main approaches: iterative (O(N)), mathematical formula (O(1)), and recursive (O(N)). The mathematical formula approach is most efficient for large values of N, while the iterative approach provides a good balance of simplicity and performance.