The Formula for the Sum of an Arithmetic Series: 1+2+3+4+5+...+n
Introduction
Have you ever wondered how to calculate the sum of a series of consecutive integers, such as 1+2+3+4+5+...+n? This formula is known as the formula for the sum of an arithmetic series, and it's an essential concept in mathematics.
The Formula
The formula for the sum of an arithmetic series is:
1 + 2 + 3 + 4 + 5 + ... + n = n(n+1)/2
Where n is the number of terms in the series.
How the Formula Works
To understand how the formula works, let's break it down:
- The formula is based on the concept of an arithmetic series, which is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term.
- The formula uses the number of terms (n) and the sum of the first and last terms (n and 1, respectively) to calculate the total sum.
- The formula can be derived by noticing that the sum of an arithmetic series can be represented as a triangular array, where each row has one more element than the previous row.
Examples
Example 1: Calculating the Sum of 1+2+3+4+5+6+7+8+9+10
Using the formula, we can calculate the sum of 1+2+3+4+5+6+7+8+9+10 as follows:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10(10+1)/2 = 55
Example 2: Calculating the Sum of 1+2+3+...+100
Using the formula, we can calculate the sum of 1+2+3+...+100 as follows:
1 + 2 + 3 + ... + 100 = 100(100+1)/2 = 5050
Conclusion
The formula for the sum of an arithmetic series is a powerful tool for calculating the sum of a series of consecutive integers. By understanding how the formula works, you can apply it to a wide range of problems and calculate the sum of any arithmetic series.
