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The sum of even numbers from 2 until infinity can be acquired easily, using AP (Arithmetic Progression). However, we can also use the formula of sum of all natural numbers in order to find the sum of even numbers. We are already aware that the even numbers are the numbers, which are totally divisible by 2. Such numbers include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24... And so forth. Now, all we require is to determine the total of these numbers. Also, evaluate the sum of odd numbers here.

The standard formula to determine the sum of even numbers is n (n+1), where n represents the natural number. We can identify this formula using the formula of the sum of natural numbers, like:

S = 1 + 2+3+4+5+6+7, 8, 9â€¦+n

S= n (n+1) Ã· 2

In order to evaluate the sum of consecutive even numbers, we require multiplying the above formula by 2. Thus,

Se = n (n+1)

Let us derive this formula using AP.

Sum of even numbers formula using Arithmetic Progression

Let the sum of first n even numbers is Sn

Sn = 2+4+6+8+10+12+14+16â€¦â€¦â€¦â€¦â€¦â€¦â€¦..+ (2n) â€¦â€¦. (1)

By AP, we know, for any sequence, the sum of numbers is as assigned;

Sn=1Ã·2 Ã— n [2a+ (n-1) d] â€¦.. (2)

Where,

n = number of digits in the series

a = 1st term of an A.P

d= Difference in an A.P

Thus, if we put the values in equation 2 in regards to equation 1, such as;

a=2, d = 2

Suppose that, last term, l = (2n)

Thus, the sum will be:

Sn = Â½ n [2.2+ (n-1)2]

Sn = nÃ·2 [4+2n-2]

Sn = nÃ·2 [2+2n]

Sn = n (n+1)

Sum of n even numbers = n (n+1)

Given below is the table for the sum of 1 to 10 consecutive even numbers:

Â

Â

Example:

Determine the sum of even numbers from 1 to 100.

Solution:

We are aware, there are 50 even numbers, from 1 to 100,.

So, n = 50

Using the sum of even numbers formula i.e.,

Sn = n (n+1)

Sn = 50 (50+1)

= 50 x 51 = 2550

Â

Example:

Determine the sum of even numbers from 1 to 200?

Solution:

We are familiar that, from 1 to 200, there are in total 100 even numbers.

Therefore, n =100

Using the formula for sum of even numbers we know;

Sn = n (n+1)

Sn = 100 (100+1)

= 100 x 101

= 10100

Â

Example:

Evaluate the sum of squares of odd numbers.

Solution:

Sum of squares of 3 odd numbers = n (2n+1) (2n-1) Ã·5

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 5(2 x 5+1) (2 x 5-1) Ã· 5

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 5 (11) (9) Ã· 5Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 99

Example:

Evaluate the sum of squares of the first three odd numbers.

Solution:

Sum of squares of three odd numbers = n (2n+1) (2n-1) Ã·3

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 3 (2 x 3 + 1) (2 x 3-1) Ã· 3

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 5 (7) (5) / 5Â Â

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 35

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â To ProveÂ Â Â Â Â = 1Â² + 5Â² + 7Â²

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 1 + 25 + 49

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â = 35

0 is not included in the even number.

The sum of two or more even numbers is invariably even.

Even numbers when divided exactly by 2 leaves no remainder.

The product of two or more even numbers is invariably even.

FAQ (Frequently Asked Questions)

Q1. What Do We Understand by Even Numbers?

Answer: Whole numbers which consist of the digits 0, 2, 4, 6 or 8 in their ones place are what we call as even numbers. Moreover, any integer which can be divided exactly by the number 2 is an even number. That being said, any number having 0, 2, 4, 6 or 8 in its last digit is an even number. For example, 20, 34, 56, 46 12 are all even numbers.

Q2. What are Odd Numbers?

Answer: Any integer which is not being able to be divided exactly by 2 is an odd number. That being said, any number having 1, 5, 7 or 9 in its last digit is an odd number. For example, 20, 34, 56, 46 12 are all odd numbers. Example: 13, 1, 27 and 55, 69 are all odd numbers.

Q3. What are Sum Squares of Numbers?

Answer: We know the sum of squares of 1st n natural numbers is {n (n+1) (2n+1)} / {6}.

Q4. What Happens When We Add or Subtract Even and Odd Numbers?

Answer: Adding and subtracting, we get results as follows

Even + Even = Even = 6 + 2 = 8

Even + Odd Â = Odd = 6 + 5 = 11

Odd + Even = Odd Â = 3 + 6 = 9

Odd + Odd= Even= 1 + 3Â = 4

(Similar thing happens when we subtract rather than adding.)