Sum of Squares of First n Natural Numbers with Formula
Sum of Squares of First n Natural Numbers is the total obtained by adding 1², 2², 3² and so on up to n². The standard formula is n(n + 1)(2n + 1) ÷ 6. This page explains the formula, calculation steps, useful shortcuts, range-based problems and common examples.
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What Is the Sum of Squares of the First n Natural Numbers?
Here, n is a positive integer and the natural numbers are taken as 1, 2, 3, ..., n. For n = 5, the sum is 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55. The same result follows from 5 × 6 × 11 ÷ 6 = 55.
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Special Sums: n, n² and n³ Formulas
Learn the formulas for the sums of the first n natural numbers, their squares, and their cubes, with a clear understanding of how these special summations are represented.
Sum of Squares of First n Natural Numbers Formula & Tricks
Important Formulas
Use this formula when the terms are the squares of all natural numbers from 1 through n.
Find the sum up to b and subtract the sum up to a − 1.
Quick Tricks
In n(n + 1)(2n + 1) ÷ 6, divide one or more factors by 2 and 3 before multiplication whenever possible. This reduces arithmetic errors.
For squares beginning at a number greater than 1, calculate the total up to the last number and subtract the total up to the number before the first term.
Sum of Squares of First n Natural Numbers Concepts
Applying the Formula for a Given n
For n = 10, the sum is 10 × 11 × 21 ÷ 6. Since 10 ÷ 2 = 5 and 21 ÷ 3 = 7, the result is 5 × 11 × 7 = 385.
Sum of Squares Over a Restricted Range
Let S(k) = k(k + 1)(2k + 1) ÷ 6. Then a² + (a + 1)² + ... + b² = S(b) − S(a − 1). This method avoids adding every square separately.
Difference Between Consecutive Square Sums
If S(n) = 1² + 2² + ... + n², then S(n) − S(n − 1) = n². This follows because the two sums contain all the same terms except n².
Related Standard Sums
The three commonly used formulas are: 1 + 2 + ... + n = n(n + 1) ÷ 2; 1² + 2² + ... + n² = n(n + 1)(2n + 1) ÷ 6; and 1³ + 2³ + ... + n³ = [n(n + 1) ÷ 2]².
Sum of Squares of First n Natural Numbers Data Table
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Small Values of the Square Sum
Values of 1² + 2² + ... + n² for the first five positive integers.
| n | Formula calculation | Sum of squares |
|---|---|---|
| 1 | 1 × 2 × 3 ÷ 6 | 1 |
| 2 | 2 × 3 × 5 ÷ 6 | 5 |
| 3 | 3 × 4 × 7 ÷ 6 | 14 |
| 4 | 4 × 5 × 9 ÷ 6 | 30 |
| 5 | 5 × 6 × 11 ÷ 6 | 55 |
Note: The sequence of square sums is 1, 5, 14, 30, 55.
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Revision Points
Key facts for the sum of squares of the first n natural numbers.
- 1² + 2² + ... + n² = n(n + 1)(2n + 1) ÷ 6.
- For a range from a to b, use S(b) − S(a − 1).
- Cancel factors with 2 and 3 before multiplying the three factors.
- S(n) − S(n − 1) = n².
- Do not confuse the square sum with n(n + 1) ÷ 2, which is the sum of natural numbers.
- For n = 0, the sum is 0; the usual natural-number formula is generally applied for positive integer n.
Sum of Squares of First n Natural Numbers FAQs
What is the sum of squares of the first n natural numbers formula?
The formula is 1² + 2² + ... + n² = n(n + 1)(2n + 1) ÷ 6.
What is the sum of the squares of the first 10 natural numbers?
Use 10 × 11 × 21 ÷ 6 = 385. Therefore, 1² + 2² + ... + 10² = 385.
What is the sum of squares from 6 to 10?
Calculate S(10) − S(5) = 385 − 55 = 330. Thus, 6² + 7² + 8² + 9² + 10² = 330.
How is the sum of squares different from the sum of the first n natural numbers?
The sum of natural numbers is n(n + 1) ÷ 2, while the sum of their squares is n(n + 1)(2n + 1) ÷ 6.
What is the sum of squares of the first 20 natural numbers?
20 × 21 × 41 ÷ 6 = 2870. Hence, the sum is 2870.
If S(n) is the sum of squares up to n, what is S(n) − S(n − 1)?
S(n) − S(n − 1) = n² because the two sums differ only by the final term n².