Sum of N Even Numbers
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 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.
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
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 leaves no remainder.
The product of two or more even numbers is invariably even.