What is the sum of squares of first n natural numbers?

The sum of squares of the first 'n' natural numbers is given by the exact formula: n(n+1)(2n+1) / 6.

Understanding the Sum of Squares

The sum of squares of the first 'n' natural numbers refers to the total value obtained by adding the squares of each positive integer from 1 up to 'n'. This can be represented mathematically as:
1² + 2² + 3² + ... + n²

This formula is a fundamental concept in mathematics and finds applications in various fields, from statistics to computer science, for efficiently calculating sums in series.

The Exact Formula

According to Cuemath, the precise formula for the sum of squares of the first 'n' natural numbers is:

$S_n = \frac{n(n+1)(2n+1)}{6}$

Where:

• $S_n$ represents the sum of the squares of the first 'n' natural numbers.
• 'n' is the count of natural numbers (positive integers) whose squares are being summed.

Example Calculation

To illustrate how the formula works, let's find the sum of squares of the first 3 natural numbers (where n=3):

First, by direct calculation:
1² + 2² + 3² = 1 + 4 + 9 = 14

Now, using the formula:

• Substitute n = 3 into the formula:
  Sum = 3 (3 + 1) (2 * 3 + 1) / 6
• Perform the operations within the parentheses:
  Sum = 3 4 (6 + 1) / 6
  Sum = 3 4 7 / 6
• Multiply the numbers in the numerator:
  Sum = 84 / 6
• Divide to get the final sum:
  Sum = 14

As demonstrated, the formula provides the exact same result as the direct summation, confirming its accuracy and efficiency, especially for larger values of 'n'.

Related Sum of Squares Formulas

The concept of summing squares can also be extended to specific sequences, such as even or odd numbers. The Cuemath reference provides distinct formulas for these variations:

These specific formulas allow for quick calculation of sums of squares for different numerical patterns, making complex series summations more manageable.