The sum of even numbers from 2 until infinity can be acquired easily, using AP (Arithmetic Progression). However, we can also use the formula of the sum of all natural numbers in order to find the sum of even numbers. We are already aware that the even numbers are the numbers, which are totally divisible by 2. Such numbers include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24... And so forth. Now, all we require is to determine the total of these numbers. Also, evaluate the sum of odd numbers here.
The memories of the students of learning the even numbers for the first time in the subject of Maths goes far back to the early days of schools. Hence, the meaning or the definition of the Even number is not necessary for the students now. Because all the students know that the numbers which are divisible by 2 are the even numbers such, 2, 8, 22, 252, 398 etc because all of these numbers can be divided by 2.
There is an interesting thing about these numbers that it is to say, even numbers, and that thing is it really easy for the students to find the sum total of progressive even numbers such as 10 + 12 + 14 + 16 and so on up to the infinity. But for that, you are required to know the formula for the arithmetic progression, and hence Vedantu has brought to the students the formula for finding the sum of the even numbers, along with its solved example.
Sum of Even Numbers Formula
The standard formula to determine the sum of even numbers is n (n+1), where n represents the natural number. We can identify this formula using the formula of the sum of natural numbers, like
S = 1 + 2+3+4+5+6+7, 8, 9…+n
S= n (n+1) ÷ 2
In order to evaluate the sum of consecutive even numbers, we require multiplying the above formula by 2. Thus,
Se = n (n+1)
Let us derive this formula using AP.
Sum of even numbers formula using Arithmetic Progression
Let the sum of first n even numbers is Sn
Sn = 2+4+6+8+10+12+14+16…………………..+ (2n) ……. (1)
By AP, we know, for any sequence, the sum of numbers is as assigned;
Sn=1÷2 × n (2a+ (n-1) d)….. (2)
n = number of digits in the series
a = 1st term of an A.P
d= Difference in an A.P
Thus, if we put the values in equation 2 in regards to equation 1, such as;
a=2, d = 2
Suppose that, last term, l = (2n)
Thus, 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 n even numbers = n (n+1)
Sum of First Ten Even Numbers
Given below is the table for the sum of 1 to 10 consecutive even numbers:
Determine the sum of even numbers from 1 to 100.
We are aware, there are 50 even numbers, from 1 to 100,.
So, n = 50
Using the sum of even numbers formula i.e.,
Sn = n (n+1)
Sn = 50 (50+1)
= 50 x 51 = 2550
Determine the sum of even numbers from 1 to 200?
We are familiar that, from 1 to 200, there are in total 100 even numbers.
Therefore, n =100
Using the formula for sum of even numbers we know;
Sn = n (n+1)
Sn = 100 (100+1)
= 100 x 101
Evaluate the sum of squares of odd numbers.
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
Evaluate the sum of squares of the first three odd numbers.
Sum of squares of three odd numbers = n (2n+1) (2n-1) ÷3
= 3 (2 x 3 + 1) (2 x 3-1) ÷ 3
= 5 (7) (5) / 5
To Prove = 1² + 5² + 7²
= 1 + 25 + 49
Benefits of Having the Formula for the Sum of Even Numbers
You may be wondering that why to add one more formula to the list of formulas that you already have to learn. Because it seems that there is no need for the formula for the sum of even numbers because we can simply add it easily. And yes, it is true that we can add it easily but what when there are so many even numbers given, then adding the complete sum either take a too long time, with the great chances of the errors, or it becomes simply impossible for us to solve.
And hence, having a formula really helps in this situation, because the formula makes it really easy for you to calculate the sum of any numbers. Also, from the perspective of the exam it is again very useful because the questions in which the students have performed all the steps in a correct manner, then and only then, the student gets the complete marks for that particular question. And also, solving the question on the sum of even numbers without the formula takes so much time, which is also not good for the exam.
Advantages of Having the Examples for the sum of Even Numbers
In the subject of Mathematics, Examples plays a vital role for the students. Because only having the formula, or even the explanation of the topic is not enough. Because both of these only tells the students what to do and how to do, that it is to say, it tells the approach to solve the question. But does not necessarily show how to do that practically.
And hence, in such situations, the examples come to help the students. Because example shows, what the normal explanation and the formula tells. That is to say, it shows how to solve the question by implementing the given formula in a step-by-step manner. By studying the examples students can know how to solve the particular question, especially various types of questions by implementing the formula.
Examples acquaint the students with all the possibilities of the questions for finding the sum of even numbers. And hence, Vedantu provides the students with detailed examples solved in a step-by-step manner.
Mastering the Topic of Sum of Even numbers
Maths is a subject which is practical in nature, and hence it requires the students to have as much practice as possible. And also, that is the best way to remember the formula for the sum of even numbers as well. By solving more questions your clarity regarding the various types of questions gets improved, and also it improves the speed of the students in solving the questions.
But most importantly it boosts the confidence of the students in solving the questions. And hence, practising the questions of the sum of Even numbers is the best way to master the topic.
0 is not included in the even number.
The sum of two or more even numbers is invariably even.
Even numbers when divided exactly by 2 leave no remainder.
The product of two or more even numbers is invariably even.