The sum of squares of the first n natural numbers helps us find the total of 1² + 2² + 3² + … + n² without calculating each square individually. This formula is widely used in mathematics, sequences, statistics, physics, and competitive exams. It simplifies long calculations and gives results quickly, especially when n is large. Understanding this formula builds a strong foundation in algebra and number series.
Sum of Squares Formula Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| S = n(n + 1)(2n + 1) / 6 | S = Sum of squares | To find the total of squared natural numbers instantly |
| n = number of terms | Used in algebra, sequences & aptitude exams |
What is the Sum of Squares in Maths?
The sum of squares means adding the squares of natural numbers one after another. For example, the sum of squares of the first 3 numbers is 1² + 2² + 3² = 14. Instead of calculating each square manually, the formula S = n(n + 1)(2n + 1)/6 makes the process much faster.
This formula is derived from rules of arithmetic sequences and is especially helpful when n is large.
Steps to apply the formula:
- Identify the value of n.
- Substitute it into the formula.
- Simplify the expression to get the sum.
It is important in exams like CUET, SSC, Banking, JEE, Railways, and is used in physics, computer science, and statistics, especially while calculating motion, energy, and variance.
Examples to Calculate Sum of Squares
Example 1: First 5 Natural Numbers
Step 1: n = 5
Step 2: S = {5(5 + 1)(2 × 5 + 1)} / 6
= (5 × 6 × 11) / 6
= 55
So, the sum of squares of the first 5 natural numbers is 55.
Example 2: First 10 Natural Numbers
Step 1: n = 10
Step 2: S = {10(10 + 1)(2 × 10 + 1)} / 6
= (10 × 11 × 21) / 6
= 385
So, the sum of squares of the first 10 natural numbers is 385.
FAQs about Sum of Squares Formula
Q1. What is the sum of squares of the first 20 natural numbers?
Using S = n(n + 1)(2n + 1)/6 → S = (20 × 21 × 41)/6 = 2870.
Q2. Who discovered this formula?
Its origins are ancient, but it is commonly linked to early number theory mathematicians.
Q3. Can this formula be used for very large n?
Yes, it works efficiently even for very large values of n.
Q4. Is this formula used in statistics?
Yes, it is used in variance, standard deviation, and regression.
Q5. Does the formula include 0?
It applies to natural numbers starting from 1. Including 0 does not change the sum.
Q6. Why is this formula important for exams?
It saves time and avoids lengthy calculations, making it useful for competitive exams.