Last updated on July 5th, 2025
The sum of even numbers can be easily calculated using the formula and arithmetic progression. Even numbers are the numbers that are evenly divisible by 2, that is, 2, 4, 6, 8, 10, … In this article, we will learn more about the sum of even numbers. The formula to find the sum is S = n(n + 1).
Even numbers are the numbers that are evenly divisible by 2; they include both positive and negative numbers. For instance, -8, -4, -2, 2, 4, 6, 8, 10, 12, … It is represented as 2n, where n is an integer. Even numbers play a crucial role in arithmetic operations, algebra, and real-world applications like coding, engineering, and financial calculations.
The sum of even numbers follows a predictable pattern and can be calculated efficiently using formulas. The sum of the first n even numbers is given by S=n(n + 1). For an arithmetic sequence of even numbers within a range, the sum can be found using S = n/2 ×(First Term+Last Term), where n is the number of terms.
Let's derive the formula for the sum of the even numbers step-by-step.
The first ten even numbers are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Thus, the sum of the first ten even numbers is: Sn = 2 + 4 + 6 + 8 + 10 +… 10 terms.
By formula Sn = n(n + 1), we have S = 10(10 + 1) = 10 × 11 = 110 (n = 10)
Also, 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 = 110
Hence, checked.
What is the Sum of Even Numbers 1 to 50?
The sum of even numbers from 1 to 50 is the result of the summation of all the even numbers in the list from 1 to 50. By the definition of even numbers, the 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).
There are 25 even numbers from 1 to 50, so n = 25.
Substitute the values in the formula Sn = n(n + 1).
Therefore, S = 25(25 + 1) = 25 × 26 = 650
What is the Sum of Even Numbers 51 to 100
The sum of even numbers from 51 to 100 is the summation of all 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, 100. Thus, there are 25 even numbers from 51 to 100.
Here, a = 52, d = 2, n = 25
Applying the sum formula,
S = ½ × 25 [(2 × 52) + (25 – 1)2]
S = ½ × 25[104 + (24)2]
S = 25/2 [152]
S = [25 × 76] = 1900
The sum of even numbers has numerous applications across various fields. Let us explore how the sum of even numbers is used in different areas:
Summing even numbers is a straightforward process that can be quickly mastered using a few key formulas and techniques. The tips and tricks mentioned below will help you solve problems:
Students tend to make mistakes when finding the sum of even numbers. Let us see some common mistakes and how to avoid them:
What is the sum of the first 5 even numbers?
30
The first 5 even numbers are: 2, 4, 6, 8, and 10.
You can add them directly:
2 + 4 + 6 + 8 + 10 = 30
Alternatively, using the formula for the first n even numbers:
Sum = n (n + 1) where n = 5.
5 × (5 + 1) = 5 × 6 = 30.
What is the sum of the first 8 even numbers?
72
Even numbers: 2, 4, 6, 8, 10, 12, 14, 16.
Use the formula:
Sum=n(n + 1) where n=8.
8×9=72.
Given a list [1,2,3,4,5,6], what is the sum of its even numbers?
12
Identify the even numbers in the list: 2, 4, and 6.
Sum them directly: 2 + 4 + 6 = 12.
What is the sum of the first 7 even numbers using the formula: n(n + 1)?
56
Set
n=7.
Apply the formula:
Sum=7(7 + 1) = 7 × 8 = 56.
What is the sum of the even numbers between 4 and 20 (inclusive)?
108
The sequence is: 4, 6, 8, …, 20.
Determine the number of terms:
Count = (20 - 4)/2 + 1
= (16/2) + 1
= 8 + 1
= 9
Apply the arithmetic series formula:
Sum = 9/2 x (4 + 20) = (9/2) × 24 = 9 × 12 = 108
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.