Sum of Even Numbers
2026-02-28 01:06 Diff
https://brightchamps.com/en-us/math/numbers/sum-of-even-numbers

We know that even numbers are those numbers that are divisible by 2. And the difference between any two consecutive numbers will be 2. If we list all the even numbers in the sequence from 1 to 100, we will get exactly 50, because every second number is even. Therefore, we can easily calculate the sum of the first 100 even numbers using standard formulas for even numbers and arithmetic progressions. 


Here, we know n = 50. 
We can substitute the value of n in the formula of the sum of even numbers, Sn = n(n + 1). 
Therefore, Sn = 50(50+1) 
= 50 x 51 
= 2550

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,


\(S = \frac{25}{2} [104 + (24 \times 2)] \)

\(= \frac{25}{2} [104 + 48] = \frac{25}{2} \times 152 = 25 \times 76 = 1900 \)