Content Curator
The sum of even numbers is the number sequence beginning with 2 and ending with infinity. Even numbers, as we all know, are those that are divided by 2, such as 2,4,6,8,10, and so on. We must apply the sum of arithmetic progression formulas or the sum of natural numbers formulas to calculate the sum of the even numbers formula.
| Table of Content |
Key Takeaways: Sum of even numbers, Even numbers, arithmetic progression, natural numbers, integers
Also read: Isosceles Triangle Theorems
What is the Sum of Even Numbers?
[Click Here for Sample Questions]
Sum of even integers from 2 to infinity which can be calculated very easily using both the Arithmetic Progressions and the sum of all natural numbers formulas are known as Sum of Even Numbers. We already know that even numbers are those that are totally divisible by two. 2, 4, 6, 8, 10, 12, 14, 16, and so on are the numbers . Now we must calculate the sum of these figures.
Sum of Even Numbers Formula
[Click Here for Sample Questions]
The sum of even numbers is calculated using the formula n(n+1), where n refers to the natural number. This formula can be found using the sum of natural numbers method, such as S = 1 + 2+3+4+5+6+7...+n
S= n(n+1)/2
We need to multiply the above formula by 2 to get the sum of successive even numbers. Hence,
Se = n(n+1)
Let's use AP to get this formula.
Also read: Bayes Theorem Formula
Sum of Even Numbers Formula Using AP
[Click Here for Sample Questions]
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 = ½ × n [2a + (n – 1) d] ……..(2)
Where,
n refers to the number of digits in the series
a refers to the First term of an A.P
d refers to the Common difference in an A.P
Hence, if we put the values in equation 2 in consider 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 First Ten Even Numbers
[Click Here for Sample Questions]
Let us look at a table for the sum of even numbers from 1 to 10 that are consecutive: Sn = 2+4+6+8+10+... 10 terms.
| Nth term in the AP | Sum of Even Numbers Formula Sn= n(n+1) | Verification of the sum |
|---|---|---|
| 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 |
Points to Remember
Following are some important points:
- The sum of even integers from 2 to infinity can be easily calculated using both Arithmetic Progression and the sum of all natural numbers formula.
- The sum of even numbers is the number sequence beginning with 2 and ending with infinity.
- Even numbers, as we all know, are those that are divided by 2, such as 2,4,6,8,10, and so on
- The sum of even numbers is calculated using the formula n(n+1), where n refers to the natural number.
Also read: Conditional Probability Formula
Sample Questions
Ques: Determine the sum of the first 20 even numbers? (2 Marks)
Ans: The natural numbers are there among the first 20 numbers. So, n = 20.
Now, let us calculate the sum of the first 20 even numbers with the help of formula,
Sn= n(n+1)
Sn= 20(20+1)
Sn= 420
Hence, the sum of the first 20 even numbers is 420.
Ques: Calculate the sum of the first 10 multiples of 8. (2 Marks)
Ans: The first 10 multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88 and n = 10. Now, let us calculate the sum of the first 10 multiples of 8 by using the arithmetic progression formula Sn=1/2×n[2a+(n-1)d]
where a = 8 and d = 8,
Sn=1/2×n[2a+(n-1)d]
Sn=1/2×10[2×8+(10-1)8]
Sn= 5 × 88
Sn=440
Hence, the sum of the first 10 multiples of 8 is 440.
Ques: Find the sum of even numbers from 1 to 200. (2 Marks)
Ans: We are already aware that there are 100 even numbers between the numbers 1 to 200. So, n = 100
Let's calculate the sum by using the formula
Sn = n(n+1)
Sn = 100(100+1)
Sn = 10,100.
Hence, the sum of the even numbers from 1 to 200 is 10,100.
Ques: Determine the sum of even numbers from 1 to 50? (2 Marks)
Ans: There are 25 even numbers from 1 to 50,
So, n = 25
With the help of formula of sum of even numbers we know;
Sn = n(n+1)
Sn = 25(25+1) = 25 x 26 = 650
Ques: Determine the sum of the first 100 even numbers? (2 Marks)
Ans: There are 50 even numbers. from 1 to 100
So, n = 50
With the help of formula of sum of even numbers we know;
Sn = n(n+1)
Sn = 50 (50+1) = 50 x 51 = 2550.
Ques: What will be the sum of all the even numbers from 41 to 80? (2 Marks)
Ans: Even number between 41 to 80
⇒ 42, 44, 46, 48, 50 ..............80
⇒ This series is in AP
Where, a = 42
⇒ d = a2 - a1
⇒ 44 - 42 = 2
Tn = a + (n-1)d
⇒ 80 = 42 + (n-1)2
⇒ 38 = (n-1)2
⇒ n = 20
⇒ Sn = 20/2{2×42+(20−1)2}
⇒ 20 {42 + 19}
⇒ 20 × 61 = 1220
Ques: Determine the sum of squares of odd numbers. (2 Marks)
Ans: Sum of squares of 3 odd numbers = n (2n + 1) (2n – 1) ÷ 5
= 5 (2 x 5 + 1) (2 x 5 – 1) ÷ 5
= 5 (11) (9) ÷ 5
= 99
Also Read:
Comments