The sum of squares of the first n even numbers is precisely given by the formula: [2n(n + 1)(2n + 1)] / 3. This formula allows for the direct calculation of the sum of the squares of any specified number of initial even integers.
Understanding the Sum of Squares of Even Numbers
When we talk about the sum of squares of even numbers, we are referring to the sum of a sequence where each term is the square of an even number. For instance, the first few even numbers are 2, 4, 6, 8, and so on. Their squares are 2²=4, 4²=16, 6²=36, 8²=64, and so forth. The sum would then be 4 + 16 + 36 + 64 + ... for a given number of terms.
The Formula for the Sum of Squares of First n Even Numbers
The formula to calculate the sum of the squares of the first n even numbers is:
$$ S_{even} = \frac{2n(n + 1)(2n + 1)}{3} $$
Where:
- $S_{even}$ represents the sum of the squares of the first n even numbers.
- n is the count of even numbers you are summing.
How to Calculate the Sum: Practical Example
Let's illustrate how to use this formula with an example.
Example: Find the sum of the squares of the first 4 even numbers.
The first 4 even numbers are 2, 4, 6, and 8.
Their squares are:
- $2^2 = 4$
- $4^2 = 16$
- $6^2 = 36$
- $8^2 = 64$
The manual sum is $4 + 16 + 36 + 64 = 120$.
Now, let's use the formula with n = 4:
-
Substitute n = 4 into the formula:
$S_{even} = \frac{2 \times 4 \times (4 + 1) \times (2 \times 4 + 1)}{3}$ -
Perform the operations within the parentheses:
$S{even} = \frac{2 \times 4 \times (5) \times (8 + 1)}{3}$
$S{even} = \frac{2 \times 4 \times 5 \times 9}{3}$ -
Multiply the numbers in the numerator:
$S{even} = \frac{8 \times 5 \times 9}{3}$
$S{even} = \frac{40 \times 9}{3}$
$S_{even} = \frac{360}{3}$ -
Divide to get the final sum:
$S_{even} = 120$
Both the manual calculation and the formula yield the same result, confirming the formula's accuracy.
Related Sum of Squares Formulas
Understanding the sum of squares for even numbers can be placed in context with other common sum of squares formulas:
| Type of Numbers | Formula |
|---|---|
| Sum of squares of first n natural numbers | [n(n+1)(2n+1)] / 6 |
| Sum of squares of first n even numbers | [2n(n + 1)(2n + 1)] / 3 |
| Sum of squares of first n odd numbers | [n(2n+1)(2n-1)] / 3 |
These formulas provide efficient methods for calculating sums of squares without needing to sum each individual squared term, which becomes particularly useful for large values of n.
