How to Find the Sum of Cubes of n Natural Numbers

The concept of sum of cubes of n natural numbers is essential in mathematics and helps in solving real-world and exam-level problems efficiently.

Understanding Sum of Cubes of n Natural Numbers

A sum of cubes of n natural numbers refers to finding the total sum when you cube each number from 1 to n and then add all those cubes together. This concept is widely used in algebra, coding, mathematical sequences, and number theory. It also appears in problems involving polynomial identities and forms the basis for several competitive exam questions.

Formula Used in Sum of Cubes of n Natural Numbers

The standard formula is: \( S = [1^3 + 2^3 + 3^3 + \dots + n^3] = \left[\dfrac{n(n+1)}{2}\right]^2 \)

Here’s a helpful table to understand the sum of cubes of n natural numbers more clearly:

Sum of Cubes Table (First Few n)

n	Sum (S = 1³ + 2³ + ... + n³)	Formula Value
1	1	(1×2/2)2 = 12 = 1
2	1 + 8 = 9	(2×3/2)2 = 32 = 9
3	1 + 8 + 27 = 36	(3×4/2)2 = 62 = 36
4	1 + 8 + 27 + 64 = 100	(4×5/2)2 = 102 = 100
5	1 + 8 + 27 + 64 + 125 = 225	(5×6/2)2 = 152 = 225

This table shows how the pattern of the sum of cubes of n natural numbers forms perfect squares, making the calculation easier for any natural number value.

Worked Example – Solving a Problem

Let us solve step-by-step how to find the sum of cubes of the first 10 natural numbers.

1. Write down the formula:

2. Substitute n = 10:

3. Calculate the value inside the bracket:

4. Square the result:

Therefore, the sum of cubes of the first 10 natural numbers is 3025.

Example: Sum of Cubes from 5 to 14

To find the sum of cubes from 5 to 14, calculate the sum up to 14 and subtract the sum up to 4:

1. Find sum for n = 14:

2. Find sum for n = 4:

3. Subtract:

So, the sum of cubes from 5 to 14 is 10,925.

Proof of Sum of Cubes of n Natural Numbers

The formula for the sum of cubes of n natural numbers can be proven using mathematical induction or polynomial identities. Here is a logical step-by-step derivation:

1. Assume \( S = 1^3 + 2^3 + 3^3 + \ldots + n^3 \).

2. Recall that \( \sum_{k=1}^{n} k^3 = [\sum_{k=1}^{n} k]^2 \).

3. But, \( \sum_{k=1}^{n} k = \dfrac{n(n+1)}{2} \).

4. Therefore,

This proof links the sum of cubes to a perfect square of the sum of natural numbers.

Sum of Cubes in Different Programming Languages

The sum of cubes formula can be coded in C, Python, or Java:

Python Example:

n = 10
S = (n * (n + 1) // 2) ** 2
print("Sum of cubes:", S)

Using Recursion in C:

int sumOfCubes(int n) {
  if (n == 1) return 1;
  else return n*n*n + sumOfCubes(n-1);
}

With these codes, students can easily calculate sum of cubes for any number. For more programming techniques, explore coding topics at Vedantu.

Practice Problems

• Find the sum of cubes of the first 8 natural numbers.
• Calculate the sum of cubes from 4 to 12.
• Write a Python code to compute the sum of cubes of first n natural numbers.
• What is the sum of cubes for n = 15?

Common Mistakes to Avoid

• Forgetting to square the result after using the sum formula.
• Mixing up the sum of cubes with the sum of squares formula.
• Not substituting the value of n correctly in the formula.

Real-World Applications

The concept of sum of cubes of n natural numbers appears in areas such as computer algorithms, data analysis, calculating volumes, banking interests, and combinatorics. Vedantu helps students see how such maths concepts are used both in exams and daily life settings.

We explored the idea of sum of cubes of n natural numbers, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.

Further Reading and Related Concepts

• Cubes and Cube Roots
• A Cube Minus B Cube Formula
• Perfect Cube of Numbers
• Algebraic Identities
• Binomial Theorem