Cube Sum of First N Natural Numbers in Java

In this article, we will learn to write a Java program to calculate the sum of cubes of the first n natural numbers.

Understanding the Cube Sum Formula

The sum of cubes of the first n natural numbers follows a mathematical formula ?

S = 1³ + 2³ + 3³ + ... + n³ = { ( n ( n + 1 ) ) / 2 }²

Different Approaches

Following are the two different approaches to printing the cube sum of first n natural numbers ?

• Using a Loop
• Using a Mathematical Formula

Using a Loop

The first approach involves iterating through the first n natural numbers, calculating the cube of each number, and adding it to a running sum.

Following are the steps to calculate the cube sum of the first n natural numbers ?

• The method first_n_nat_no takes an integer val as input, representing the number of natural numbers.
  
• A for loop iterates from 1 to val, calculating the cube of each number (x * x * x) and adding it to ini_sum.
• The final sum is returned and printed in the main method.

for (int x = 1; x <= val; x++) {
            ini_sum += x * x * x; // Cube of x and add to sum     
}

Example

Following is the Java code to cube the sum of the first n natural numbers ?

import java.util.*;
import java.lang.*;

public class Demo {
    public static int first_n_nat_no(int val) {
        int ini_sum = 0;
        for (int x = 1; x <= val; x++) {
            ini_sum += x * x * x; // Cube of x and add to sum
        }
        return ini_sum;
    }

    public static void main(String[] args) {
        int val = 7; 
        System.out.println("The sum of cube of first 7 natural numbers is ");
        System.out.println(first_n_nat_no(val));
    }
}

Output

The sum of cube of first 7 natural numbers is
784

Time Complexity: O(n), time since it iterates through the numbers from 1 to n.
Space Complexity: O(1), as we only use a few integer variables.

Using the Mathematical Formula

The second approach uses a mathematical formula to calculate the sum of cubes of the first n natural numbers. The formula is:

The sum of cubes= { ( n ( n + 1 ) ) / 2 } ²

This formula is derived from the fact that the sum of cubes of the first n natural numbers is equal to the square of the sum of the first n natural numbers.

int sum = (val * (val + 1)) / 2; // Sum of first n natural numbers
return sum * sum; // Square of the sum

Example

Following is the Java code to cube the sum of the first n natural numbers ?

import java.util.*;
import java.lang.*;

public class Demo {
    // Method to calculate the sum of cubes using the formula
    public static int first_n_nat_no_formula(int val) {
        int sum = (val * (val + 1)) / 2; // Sum of first n natural numbers
        return sum * sum; // Square of the sum
    }

    public static void main(String[] args) {
        int val = 7; // Number of natural numbers
        System.out.println("The sum of cube of first 7 natural numbers is ");
        System.out.println(first_n_nat_no_formula(val));
    }
}

Output

The sum of cube of first 7 natural numbers is
784

Time Complexity: O(1), (constant time) since it performs a fixed number of operations.
Space Complexity: O(1), as it uses only integer variables.

Conclusion

Both approaches are effective for calculating the sum of cubes of the first n natural numbers. The loop approach is straightforward and works well for small inputs, while the mathematical formula approach is highly efficient and recommended for larger inputs.