Sum of Even Numbers

Sum of Even Numbers

Calculating the Sum of Even Numbers is simple using Arithmetic Progression and the formula for the sum of all natural numbers. We already know that even numbers are those that are divisible by two, from 2 to infinity, including 2, 4, 6, 8, 10, 12, 14, 16, etc. Now, let’s find the sum total of these numbers. Se = n(n+1) is the formula to get the sum of even numbers.

In this post, we will study the formula for the sum of even numbers and how to compute the sum of even numbers using worked examples.

What is the Sum of Even Numbers?

The sum of even numbers from 2 to infinity can be simply calculated using arithmetic progression since the set of even numbers is itself an arithmetic progression with a defined difference between each succeeding term. Using the formula for the sum of natural numbers, such as S = 1+2+3+4+5+6…+n, the formula for the sum of even numbers may be determined. Thus, S= n(n+1)/2. To determine the sum of successive even integers, multiply the sum of the formula for natural numbers by 2. Hence, Se = n(n+1)

Sum of Even Numbers Formula

Let’s derive the sum of even numbers formula using a step-by-step procedure.

Let the sum of the first n even numbers is Sn. Thus, Sn = 2+4+6+8+10+…………………..+(2n) ……. (1)

For an arithmetic sequence, the sum of numbers is given by Sn=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 Sn = ½ n[2.2+(n-1)2] ⇒ Sn = n/2[4+2n-2] ⇒ Sn = n/2[2+2n] ⇒ Sn = n(n+1)

Therefore, the sum of n even numbers = n(n+1) or Se = n(n+1).

Sum of First Ten Even Numbers

Let’s locate the initial ten even numbers. The initial list of even numbers will consist of the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20.

Thus, the sum of consecutive even numbers from 1 to 10 is Sn = 2+4+6+8+10+… 10 terms.

According to the formula Sn = n(n+1), S = 10(10+1) = 10 x 11 = 110 (n = 10)

Also, 2+4+6+8+10+12+14+16+18+20=110

As a result, this has been examined.

Sum of Even Numbers 1 to 100

We know that even numbers are those that can be divided by two. Additionally, we are aware that the difference between any two consecutive even numbers is 2. The total of even numbers from 1 to 100 is equal to the sum of all even numbers from 1 to 100. According to the definition of even numbers, from 1 to 100 there are 50 even numbers. Thus, n = 50

Substitute the value of n in the sum of even numbers’ formula, Sn = n(n+1).

Sn = 50(50+1) = 50 x 51 = 2550

Sum of Even Numbers 1 to 50

The total of even numbers from 1 to 50 is equal to the sum of all even numbers from 1 to 50. 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, and 50, according to the definition of even numbers. Thus, n = 25.

Replace the variables in the equation Sn = n(n+1).

In conclusion, S = 25(25+1) = 25 x 26 = 650

Sum of Even Numbers 51 to 100

The sum of even numbers from 51 to 100 is equal to the sum of all even numbers from 51 to 100. The even numbers from 51 to 100 are 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, and 100. Therefore, there are 25 even numbers between 51 and 100.

Here, a = 52, d = 2, n = 25

Applying the sum of ap formula,

Sn=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

Sum of Even Numbers Formula Using AP

Sn = 2+4+6+8+10+…………………..+(2n) ……. (1)

By Arithmetic Progression, we know, for any sequence, the sum of numbers is given by;

Sn=1/2×n[2a+(n-1)d] ……..(2)

Where,

n = number of digits in the series

a = First term of an A.P

d= Common difference in an A.P

Therefore, if we put the values in equation 2 with respect to equation 1, such as;

a=2 , d = 2

Let, last term, l = (2n)

So, the sum will be:

Sn = ½ n[2.2+(n-1)2]

Sn = n/2[4+2n-2]

Sn = n/2[2+2n]

Sn = n(n+1)

⁕ Related Articles

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

Examples of Sum of Even Numbers

Question 1: What is the sum of even numbers from 1 to 50?

Solution: We know that, from 1 to 50, there are 25 even numbers.

Thus, n = 25

By the formula of the sum of even numbers we know;

Sn = n(n+1)

Sn = 25(25+1) = 25 x 26 = 650

Question 2: What is the sum of the first 100 even numbers?

Solution: We know that, from 1 to 100, there are 50 even numbers.

Thus, n = 50

By the formula of the sum of even numbers we know;

Sn = n(n+1)

Sn = 50(50+1) = 50 x 51 = 2550

Question 3: Find the sum of even numbers from 1 to 200?

Solution: We know that, from 1 to 200, there are 100 even numbers.

Thus, n =100

By the formula of the sum of even numbers we know;

Sn = n(n+1)

Sn = 100(100+1) = 100 x 101 = 10100

Example 4: 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

Se = n(n+1)

Sn= 20(20+1)

Sn= 420

Therefore, the sum of the first 40 even numbers is 420.

FAQs on Sum of Even Numbers