The formula for calculating Sum of Cubes of First n Natural Numbers helps us quickly find the total of 1³ + 2³ + 3³ + … + n³ without adding each cube one by one.
The formula is: S = [n(n + 1) / 2]²
Where:
- S = Sum of cubes of first n numbers
- n = Number of terms
Formula for Calculating 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 | To calculate total of cubic numbers |
| n = number of terms | Used in algebra, sequences, and aptitude exams |
What is Sum of Cubes in Maths?
The sum of cubes means adding the cubes of natural numbers.
For example: 1³ + 2³ + 3³ = 36.
It shows the total of cube values and often appears in algebra, series, and problem-solving.
The formula for cubes is special because it connects directly with the formula for the sum of natural numbers. In fact, the sum of cubes of the first n natural numbers is equal to the square of the sum of first n natural numbers.
Steps to apply this formula:
- Find n (the total natural numbers).
- Apply the formula S = [n(n + 1)/2]².
- Simplify step by step.
This formula is very useful in exams like CUET, SSC, Banking, JEE, and others. It also appears in real-life calculations involving volume, arrangements, or when dealing with cubic growth patterns.
Solved Examples
Example 1:
Find the sum of cubes of the first 5 natural numbers.
Solution :
Step 1: n = 5
Here, we use : S = [n(n + 1) / 2]²
Step 2: S = [5(5 + 1)/2]²
= [5 × 6 / 2]²
= [15]²
= 225
So, the sum of cubes of the first 5 natural numbers is 225
Example 2:
Find the sum of cubes of the first 10 natural numbers.
Solution :
Step 1: n = 10
Here, we use : S = [n(n + 1) / 2]²
Step 2: S = [10(11)/2]²
= [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 first 20 natural numbers?
Using the formula, S = [20(21)/2]²
= [210]²
= 44100.
Q2. How is the cube sum formula different from square sum?
The cube sum is squared from the natural number sum, while square sum has its own direct formula.
Q3. Can we use this formula for large n values?
Yes, it is perfect for large n as it avoids lengthy cubic additions.
Q4. Is this formula important for exams?
Yes, it frequently appears in aptitude and algebra questions in CUET, SSC, Banking, JEE, Railways.
Q5. Who first discovered this relationship?
The identity was known since ancient times, but mathematicians like Fermat and Gauss popularized it.
Q6. Does this formula apply if n = 0?
Yes, if n = 0, the sum is simply 0.