The sum of even numbers from 2 to infinity can be obtained easily, using Arithmetic Progression as well as using the formula of sum of all natural numbers. We know that the even numbers are the numbers, which are completely divisible by 2. They are 2, 4, 6, 8,10, 12,14, 16 and so on. Now, we need to find the total of these numbers. Also, find sum of odd numbers here. Learn about even numbers here. Basically, the formula to find the sum of even numbers is n(n+1), where n is the natural number. We can find this formula using the formula of the sum of natural numbers, such as: S = 1 + 2+3+4+5+6+7…+n S= n(n+1)/2 To find the sum of consecutive even numbers, we need to multiply the above formula by 2. Hence, Se = n(n+1) Let us derive this formula using AP. Let the sum of first n even numbers is Sn 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)Sum of Even Numbers Formula Using AP
Sum of n even numbers = n(n+1)
Sum of First Ten Even numbers
Below is the table for the sum of 1 to 10 consecutive even numbers.
| Number of consecutive even numbers (n) | Sum of even numbers (Sn = n (n+1)) | Recheck |
| 1 | 1(1+1)=1×2=2 | 2 |
| 2 | 2(2+1) = 2×3 = 6 | 2+4 = 6 |
| 3 | 3(3+1)=3×4 = 12 | 2+4+6 = 12 |
| 4 | 4(4+1) = 4 x 5 = 20 | 2+4+6+8=20 |
| 5 | 5(5+1) = 5 x 6 = 30 | 2+4+6+8+10 = 30 |
| 6 | 6(6+1) = 6 x 7 = 42 | 2+4+6+8+10+12 = 42 |
| 7 | 7(7+1) = 7×8 = 56 | 2+4+6+8+10+12+14 = 56 |
| 8 | 8(8+1) = 8 x 9 = 72 | 2+4+6+8+10+12+14+16=72 |
| 9 | 9(9+1) = 9 x 10 = 90 | 2+4+6+8+10+12+14+16+18=90 |
| 10 | 10(10+1) = 10 x 11 =110 | 2+4+6+8+10+12+14+16+18+20=110 |
Also, read:
- Sum Of N Terms
- Sum Of Squares
Solved Examples
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 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 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
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All the even and odd numbers between 1 and 100 are discussed here.
What are the even numbers from 1 to 100?
The even numbers from 1 to 100 are:
2 4 6 8 1012 14 16 18 2022 24 26 28 3032 34 36 38 4042 44 46 48 5052 54 56 58 6062 64 66 68 7072 74 76 78 8082 84 86 88 9092 94 96 98 100
What are the odd numbers from 1 to 100?
The odd numbers from 1 to 100 are:
1 3 5 7 911 13 15 17 1921 23 25 27 2931 33 35 37 3941 43 45 47 4951 53 55 57 5961 63 65 67 6971 73 75 77 7981 83 85 87 8991 93 95 97 99
Even and Odd Numbers
1. (i) Every number coming after an even number is an odd number12 + 1 = 13 16+1 = 17 176 + 1 = 177 216 + 1 = 217
(ii) Every number coming after an odd number is an even number15 + 1 = 16 19 + 1 = 20 205 + 1 = 206
419 + 1 = 420
2. Every number having 2, 4, 6, 8 and 0 at its one’s place is divisible by 2 and hence, an even number.
3. No number having 3, 5, 7 and 9 at its one's place is divisible by 2 and hence, an odd number.
4. All the even and odd numbers are whole numbers.
Related Concept
● Factors and Multiples by using Multiplication Facts
● Factors and Multiples by using Division Facts
● Multiples
● Properties of Multiples
● Examples on Multiples
● Factors
● Factor Tree Method
● Properties of Factors
● Examples on Factors
● Even and Odd Numbers
● Even and Odd Numbers Between 1 and 100
● Examples on Even and Odd Numbers
4th Grade Math Activities
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