The sum of cubes formula helps you calculate the total of numbers like 1³ + 2³ + 3³ + … + n³ quickly, without adding each cube individually. It turns a long addition into a simple mathematical expression. This concept is widely used in algebra, sequences, competitive exams, and problem-solving. Understanding this formula also reveals a beautiful relationship between cubes and natural numbers.
Formula for Sum of Cubes of First n Natural Numbers – Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| S = [n(n + 1) / 2]² | S = Sum of cubes, n = number of terms | To calculate the total of cubic numbers in algebra, sequences, and exams |
What is the Sum of Cubes in Maths?
The sum of cubes means adding the cubes of natural numbers. For example, 1³ + 2³ + 3³ = 36. It appears in algebra, number series, and many aptitude questions. What makes this formula special is that it connects directly to the formula for the sum of natural numbers. The identity states that the sum of cubes of first n natural numbers equals the square of the sum of first n natural numbers.
To apply this formula:
- Identify n, the total number of natural numbers.
- Use S = [n(n + 1)/2]².
- Simplify step by step to get the final answer.
This relationship is very useful in exams like CUET, SSC, Banking, JEE, Railways, and school-level maths. It also appears in real-life calculations involving patterns, volume growth, and number arrangements.
Examples to Calculate Sum of Cubes of First n Natural Numbers
Example 1: n = 5
Step 1: n = 5
Step 2: S = [5(5 + 1)/2]²
Step 3: S = [5 × 6 / 2]²
Step 4: S = [15]² = 225
So, the sum of cubes of the first 5 natural numbers is 225.
Example 2: n = 10
Step 1: n = 10
Step 2: S = [10(10 + 1)/2]²
Step 3: S = [10 × 11 / 2]²
Step 4: S = [55]² = 3025
So, the sum of cubes of the first 10 natural numbers is 3025.
FAQs about Sum of Cubes of First n Natural Numbers
Q1. What is the sum of cubes of the first 20 natural numbers?
S = [20 × 21 / 2]² = [210]² = 44100.
Q2. How is the cube sum formula different from square sum?
The cube sum equals the square of the natural number sum, while square sum follows a different direct formula.
Q3. Can we use this formula for large values of n?
Yes, it works perfectly for large n and saves time by avoiding long cube additions.
Q4. Is this formula important for competitive exams?
Yes, it is commonly asked in CUET, SSC, Banking, JEE, Railways, and aptitude tests.
Q5. Who discovered this cube-sum relationship?
The identity has ancient roots but became popular through mathematicians like Fermat and Gauss.
Q6. Does this formula work for n = 0?
Yes, if n = 0, the sum is 0.