Formula for 2+4+6+8+...+n
The formula for 2+4+6+8+...+n is a well-known mathematical formula that represents the sum of consecutive even numbers starting from 2 up to n. In this article, we will explore the formula, its derivation, and some examples to illustrate its usage.
Derivation of the Formula
The formula for 2+4+6+8+...+n can be derived using the concept of arithmetic series. An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term. In this case, the fixed constant is 2, and the sequence starts from 2.
Let's denote the sum of the first n even numbers as S_n. We can write S_n as:
S_n = 2 + 4 + 6 + ... + n
We can observe that each term in the sequence is 2 more than the previous term. Therefore, we can rewrite the sequence as:
S_n = 2 + (2 + 2) + (2 + 2 + 2) + ... + (2 + 2 + 2 + ... + 2)
Notice that the number of 2's in each term is increasing by 1 in each subsequent term. Therefore, we can rewrite the sequence as:
S_n = 2 + 2(2) + 2(3) + 2(4) + ... + 2(n)
Now, we can factor out the 2 from each term and rewrite the sequence as:
S_n = 2(1 + 2 + 3 + 4 + ... + n)
The sum of the first n positive integers is given by the formula:
1 + 2 + 3 + 4 + ... + n = n(n + 1)/2
Substituting this formula into the previous equation, we get:
S_n = 2(n(n + 1)/2)
Simplifying the expression, we get:
S_n = n(n + 1)
This is the formula for the sum of the first n even numbers.
Examples
Example 1
Find the sum of the first 5 even numbers.
Using the formula, we get:
S_5 = 5(5 + 1) = 30
Therefore, the sum of the first 5 even numbers is 30.
Example 2
Find the sum of the first 10 even numbers.
Using the formula, we get:
S_10 = 10(10 + 1) = 110
Therefore, the sum of the first 10 even numbers is 110.
Conclusion
In conclusion, the formula for 2+4+6+8+...+n is a powerful tool for calculating the sum of consecutive even numbers. The formula is derived using the concept of arithmetic series and can be used to find the sum of any number of consecutive even numbers.
