Calculating the Sum of an Arithmetic Series: A Step-by-Step Guide

Understanding how to find the sum of an arithmetic series is a fundamental concept in mathematics. This guide will walk you through the process of calculating the sum of an arithmetic series, using an example with a first term of 15, a common difference of 5, and a last term of 115. By the end of this article, you will be able to apply the Sum of Arithmetic Series Formula to other similar problems.

Introduction to the Arithmetic Series and Its Components

Before we dive into the calculations, let's define some key terms:

First Term (a1): The initial term of the series, which in this case is 15. Last Term (an): The final term of the series, which is 115. Common Difference (d): The constant difference between consecutive terms, which is 5. Total Number of Terms (n): The number of terms in the series.

Understanding the Arithmetic Series Formula

The formula for the nth term of an arithmetic series is given as:

an a1 (n-1)d

In our example, we can write:

115 15 (n-1)5

Calculating the Total Number of Terms (n)

To find the number of terms (n), we rearrange the equation:

115-15 (n-1)5

100/5 n-1

20 n-1

n 21

Calculating the Sum of the Arithmetic Series

Now that we know there are 21 terms, we can use the formula for the sum of an arithmetic series:

Sn (n/2)(a1 an)

Substituting the known values:

S21 (21/2)(15 115)

S21 (21/2)(130)

S21 1365

Conclusion

In conclusion, the sum of the arithmetic series with the first term 15, common difference of 5, and last term of 115 is 1365. This method can be applied to any arithmetic series by substituting the appropriate values into the formula:

Numerical values for the first term, last term, and common difference. Numerical values obtained from the given arithmetic series.

Understanding and mastering the arithmetic series and its application can greatly enhance your mathematical skills and problem-solving abilities.