Sum of Even Numbers
The sum of even numbers can be calculated easily, using Arithmetic Progression as well as using the formula of the sum of all natural numbers. We already know that the even numbers are the numbers, which are divisible by 2, starting from 2 till infinity, such as 2, 4, 6, 8,10, 12,14, 16, and so on. Now, let's find the sum total of these numbers. The formula to find the sum of even numbers is given as S_{e} = n(n+1).
In this article, let's learn about the sum of even numbers formula and how to calculate the sum of even numbers with solved examples.
1.  What is Sum of Even Numbers? 
2.  Sum of Even Numbers Formula 
3.  Sum of First Ten Even Numbers 
4.  Sum of Even Numbers 1 to 100 
5.  FAQs on Sum of Even Numbers 
What is Sum of Even Numbers?
The sum of even numbers from 2 to infinity can be easily found, using arithmetic progression as the set of even numbers is also an arithmetic progression with a fixed difference between any two consecutive terms. The formula to find the sum of even numbers can be derived using the formula of the sum of natural numbers, such as S = 1+2+3+4+5+6+7…+n. Thus, S= n(n+1)/2. Now, to find the sum of consecutive even numbers, multiply the sum of the natural number formula by 2. Hence, S_{e} = n(n+1)
Sum of Even Numbers Formula
Let's derive the sum of even numbers formula using a stepbystep procedure.
 Let the sum of first n even numbers is S_{n}. Thus, S_{n} = 2+4+6+8+10+…………………..+(2n) ……. (1)
 For an arithmetic sequence, the sum of numbers is given by S_{n}=1/2×n[2a+(n1)d] ……..(2) (where, n = number of digits in the series, a = First term of an A.P and d= Common difference in an A.P)
 Substitute the values in equation 2 with respect to equation 1. Thus, a=2 , d = 2 and let, last term, l = (2n).
 So, the sum will be S_{n} = ½ n[2.2+(n1)2] ⇒ S_{n} = n/2[4+2n2] ⇒ S_{n} = n/2[2+2n] ⇒ S_{n} = n(n+1)
Therefore, the sum of n even numbers = n(n+1) or S_{e} = n(n+1).
Sum of First Ten Even Numbers
Let us look for the first ten even numbers. The list of first even numbers will include the following even numbers  2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Thus, the sum of even numbers from 1 to 10 that are consecutive: S_{n} = 2+4+6+8+10+... 10 terms.
By formula S_{n} = n(n+1), we have S = 10(10+1) = 10 x 11 =110 (n = 10)
Also, 2+4+6+8+10+12+14+16+18+20=110
Hence checked.
Sum of Even Numbers 1 to 100
We know that even numbers are the numbers that are divisible by 2. We also know that the difference between any two consecutive even numbers is 2. The sum of even numbers from 1 to 100 will give the summation of all the even numbers in the list from 1 to 100. By the definition of even numbers, there are 50 even numbers from 1 to 100. Thus, n = 50
Substitute the value of n in the formula of the sum of even numbers, S_{n} = n(n+1)
Therefore, S_{n} = 50(50+1) = 50 x 51 = 2550
Sum of Even Numbers 1 to 50
The sum of even numbers from 1 to 50 will give the summation of all the even numbers in the list from 1 to 50. By the definition of even numbers, there even numbers from 1 to 50 include 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). Thus, n = 25.
Substitute the values in the formula S_{n} = n(n+1).
Therefore, S = 25(25+1) = 25 x 26 = 650
Sum of Even Numbers 51 to 100
The sum of even numbers from 51 to 100 will give the summation of all the even numbers in the list from 51 to 100. By the definition of even numbers, the even numbers from 51 to 100 include 52, 54, 56, 58, 60, 62, 64, 66, 68, 70,72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100. Thus, there are 25 even numbers from 51 to 100.
Here, a = 52, d = 2, n = 25
Applying the sum of ap formula,
S_{n}=1/2×n[2a+(n1)d]
S=1/2×25[2.52+(251)2]
S=1/2×25[104+(24)2]
S=25/2[152]
S=[25(76)] = 1900
Related Articles
Examples on Sum of Even Numbers

Example 1: Determine the sum of even numbers from 1 to 101.
Solution:
The even numbers from 1 to 101 are
Applying the sum of even numbers formula,
S_{e} = n(n+1)
Therefore, the sum of all even numbers from 1 to 101 is 2550.

Example 2: What is the sum of the first 20 even numbers?
Solution:
Here, n = 20.
Now, let us find the sum of the first 40 even numbers
S_{e }= n(n+1)
S_{n}= 20(20+1)
S_{n}= 420
Therefore, the sum of the first 40 even numbers is 420.
FAQs on Sum of Even Numbers
What Is the Sum of Even Numbers Formula?
The sum of even numbers formula gives the sum total of all the even numbers. The formula to find the sum of even numbers is n(n+1), where n is the natural number. This formula is derived using the formula of the sum of natural numbers.
How To Find Sum of Even Numbers?
The formula to find the sum of even numbers is n(n+1), where n is the natural number. Determine the value of 'n' from the given list and substitute its value in the sum of even numbers formula.
How To Prove Sum of Even Numbers is Even?
Even numbers are the numbers that are divisible by 2, thus we can say that even numbers are the factors of 2 and so the sum of two even numbers will have 2 as a factor. Therefore, we can conclude that the sum of even numbers is an even number.
What Is the Sum of Even Numbers Less Than 30?
The list of even numbers less than 30 includes 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, and 28. That means there are 14 even numbers that are less than 30. Therefore, the sum of even numbers less than 30 is 14(14 + 1) = 210.
What Is the Sum of Even Numbers 1 to 1000?
The sum of even numbers 1 to 1000 can be calculated as n(n + 1). Substituting the value of n(n = 500), we have the sum of even numbers 1 to 1000 = 500(500 + 1) = 250500.
What Is the Sum of Even Numbers 2 to 100?
There are 50 even numbers from 2 to 100, thus, the sum of even numbers from 2 to 100 is 50(50+1) = 2550.
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