Sum of Even Numbers Formula

The sum of even numbers is the numbers starting from 2 that goes till infinity. As we already know that even numbers are those numbers that are divisible by the number 2, for example, 2,4,6,8,10 and so on. To determine the sum of even numbers formula, we need to use the sum of arithmetic progression formula or the sum of natural numbers formula. Let us learn about the formula and solve a few examples in this section. 

What Is Sum of Even Numbers Formula?

The sum of even numbers formula is determined by using the formula to find the arithmetic progression. The sum of even numbers goes on until infinity. The sum of even numbers formula can also be evaluated using the sum of natural numbers formula. We need to obtain the formula for 2 + 4+ 6+ 8+ 10 +...... 2n. The sum of even numbers = 2(1 + 2+ 3+ .....n). This implies 2(sum of n natural numbers) = 2[n(n+1)]/2 = n(n+1)

Sum of Even Numbers Formula 

S = n(n+1) , where n is the number of terms in the series

Derivation of Sum of Even Numbers Formula 

Let us derive the sum of even numbers formula by using arithmetic progression. Let the sum of first n even numbers is S_n, so S_n = 2+4+6+8+10+…………………..+(2n) ……. (1)

By Arithmetic Progression(AP), we know, for any sequence, the sum of n terms of an AP is given by: S_n= (1/2)× n[2a+(n-1)d] ……..(2)

Where, 

n = number of terms in the series

a = First term of an arithmetic progression

d= Common difference in an arithmetic progression

Therefore, if we put the values in equation 2 with respect to equation 1, such as a = 2, d = 2, and suppose last term, l = (2n)

So, the sum will be:

S_n = ½ n[2×2+(n-1)2]

S_n = ½ n[4+2n-2]

S_n = ½ n[2+2n]

S_n = ½ 2n(n+1)

S_n = n(n+1)

Hence, the sum of even numbers formula = n(n+1)

Sum of First Ten Even Numbers 

Let us look at a table for the sum of even numbers from 1 to 10 that are consecutive: S_n = 2+4+6+8+10+... 10 terms

Example Using Sum of Even Numbers Formula

Example 1: What is the sum of the first 20 even numbers?

Solution: The natural numbers present among the first are 20 numbers. So, n = 20. 

Now, let us find the sum of the first 20 even numbers

S_n= n(n+1)

S_n= 20(20+1)

S_n= 420

Therefore, the sum of the first 20 even numbers is 420

Example 2: Find the sum of the first 10 multiples of 8

Solution: The first 10 multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88 and n = 10 

Now, let us find the sum of the first 10 multiples of 8 by using the arithmetic progression formula S_n=1/2×n[2a+(n-1)d] where a = 8 and d = 8

S_n=1/2×n[2a+(n-1)d]

S_n=1/2×10[2×8+(10-1)8]

S_n= 5 × 88

S_n=440

Therefore, the sum of the first 10 multiples of 8 is 440

Example 3: Determine the sum of even numbers from 1 to 200. 

Solution: We know that are 100 even numbers between the numbers 1 to 200. So, n = 100

Let's find the sum by using the formula

S_n = n(n+1) 

S_n = 100(100+1) 

S_n = 10,100

Therefore, the sum of the even numbers from 1 to 200 is 10,100