## Summation of Even Natural Numbers

### Summation of Even Natural Numbers

The summation of even natural numbers is a simple mathematical concept that involves adding up all even numbers from 2 to a specified number. In this post, we will explore the formula used to calculate the summation of even natural numbers, provide some examples, and discuss its applications.

Formula for Summation of Even Natural Numbers

The formula to calculate the summation of even natural numbers is:

2 + 4 + 6 + … + n = n(n+2)/2

where n is the last even number to be added.

Example 1: Find the Sum of Even Numbers from 2 to 10

To find the sum of even numbers from 2 to 10, we can use the formula mentioned above.

2 + 4 + 6 + 8 + 10 = 30

Using the formula:

n = 10, so n + 2 = 12

Therefore, the sum of even numbers from 2 to 10 is (10 x 12)/2 = 60/2 = 30.

Example 2: Find the Sum of Even Numbers from 2 to 100

To find the sum of even numbers from 2 to 100, we can use the formula mentioned above.

2 + 4 + 6 + … + 98 + 100 = ?

Using the formula:

n = 100, so n + 2 = 102

Therefore, the sum of even numbers from 2 to 100 is (100 x 102)/2 = 5100.

Applications of Summation of Even Natural Numbers

The summation of even natural numbers has various applications in mathematics and computer science. Some of them are:

- Finding the average of even numbers

We can use the summation of even natural numbers to find the average of even numbers. For example, to find the average of even numbers from 2 to 10, we can divide the sum of even numbers by the number of even numbers.

Average of even numbers from 2 to 10 = (2 + 4 + 6 + 8 + 10)/5 = 6.

- Calculation of probabilities

The summation of even natural numbers is used in the calculation of probabilities. For example, if we have two dice, we can calculate the probability of getting an even number by dividing the sum of even numbers by the total number of possible outcomes.

Probability of getting an even number on a dice = (2 + 4 + 6)/6 = 1/2.

- Calculation of work done

In physics, we can use the summation of even natural numbers to calculate the work done when a force is applied to an object. For example, if a force of 10 Newtons is applied to an object and it moves 10 meters, we can calculate the work done using the formula:

Work = Force x Distance

Work = 10 x (2 + 4 + 6 + … + 18 + 20)

Using the formula, we get:

n = 20, so n + 2 = 22

Therefore, the work done is 1100 Joules.

Here are some examples of summation of even natural numbers:

Find the sum of the first 10 even numbers: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110

Find the sum of the first 20 even numbers: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 = 420

Find the sum of the first 15 even numbers: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 = 240

Find the sum of the first 25 even numbers: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 = 650

Find the sum of the first 30 even numbers: 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 + 52 + 54 + 56 + 58 + 60 = 930

In conclusion, the summation of even natural numbers is a simple but important mathematical concept that has various applications in different fields. The formula mentioned above can be used to find the sum of even numbers from 2 to a specified number. By understanding this formula, we can apply it to different problems and make calculations much easier.