What is the sum of squares of first n natural numbers formula?
The formula for calculating Sum of Squares of First n Natural Numbers is used to find the total of 1² + 2² + 3² + … + n² without adding each square separately.
The formula is:
S = n(n + 1)(2n + 1) / 6
Where:
- S = Sum of squares of first n numbers
- n = Number of terms
Squares of first n natural numbers formula Overview
| Formula | Variables | When it is Used |
|---|---|---|
| S = n(n + 1)(2n + 1)/6 | S = Sum of squares | To find total of squared numbers quickly |
| n = number of terms | Used in algebra, sequences, and aptitude exams |
What is Sum of Squares in Maths?
The sum of squares means adding the squares of natural numbers.
Example: 1² + 2² + 3² = 14. It is often used in sequences, statistics, and problem-solving.
Instead of squaring and adding numbers one by one,
we use the formula S = n(n + 1)(2n + 1)/6.
This formula comes from arithmetic progression rules and is very useful for large values of n.
Steps to apply:
- Identify the value of n (the total natural numbers).
- Put the value in the formula.
- Simplify step by step to get the sum.
This is important in CUET, SSC, Banking, JEE, Railways, and other aptitude tests. It also has real-life applications in areas like physics (motion, energy), computer science, and statistics.
Solved Examples
Example 1:
Find the sum of squares of the first 5 natural numbers.
Solution :
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:
Find the sum of squares of the first 10 natural numbers.
Solution :
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 5 natural numbers is 385
FAQs
Q1. What is the sum of squares of first 20 natural numbers?
Using the formula, S = n(n + 1)(2n + 1)/6
S = {20(21)(41)}/6
= 2870.
Q2. Who discovered this formula?
This formula has roots in ancient mathematics but is widely attributed to work in number theory by early mathematicians.
Q3. Can this formula be used for large n values?
Yes, the formula is designed to handle even very large numbers quickly.
Q4. Is this formula used in statistics?
Yes, the sum of squares is important in variance, standard deviation, and regression analysis.
Q5. Is 0 included in this formula?
No, it applies to natural numbers starting from 1. If 0 is included, it does not affect the sum.
Q6. Why is this formula important for exams?
Because it saves time and avoids long calculations, making it very useful in aptitude and competitive exams.