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.

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)

Let's derive the sum of even numbers formula using a step-by-step 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+(n-1)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+(n-1)2] ⇒ S_n = n/2[4+2n-2] ⇒ 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).

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.

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+(n-1)d]

S=1/2×25[2.52+(25-1)2]

S=1/2×25[104+(24)2]

S=25/2[152]

S=[25(76)] = 1900

Related Articles

• Odd Numbers
• Consecutive Numbers
• Sum of n terms of an Ap
• Geometric Progressions

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.