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Welcome to our Math lesson on An Alternative Formula for the Calculation of Sn in Arithmetic Series, this is the third lesson of our suite of math lessons covering the topic of Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series., you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
How to calculate Sn in Arithmetic Series using an Alternative Formula
In the previous tutorial, we explained how to find the general term of an arithmetic sequence when the first term x1 and the common difference d are known. The formula used for this, was
xn = x1 + (n - 1) ∙ d
Substituting this expression for xn in the Gauss formula discussed in lesson 12.2.2 yields
Sn = (x1 + xn ) ∙ n/2
= [x1 + x1 + (n - 1) ∙ d] ∙ n/2
Hence, we obtain
Sn = [2x1 + (n - 1) ∙ d] ∙ n/2
This new formula doesn't require knowing the last (nth) term of a given arithmetic series but only the first term, the number of terms and the common difference.
Example 4
The first term of an arithmetic sequence is 14, the common difference between two consecutive terms is 9 and there are 23 terms in total. Calculate:
- The sum of all terms of this sequence
- The last term of this sequence
Solution 4
- Using the alternative formula for calculating the first n terms of the given arithmetic sequence yields
Sn = [2x1 + (n - 1) ∙ d] ∙ n/2
= [2 ∙ 14 + (23 - 1) ∙ 9] ∙ 23/2
= [28 + 22 ∙ 9] ∙ 23/2
= (28 + 198) ∙ 23/2
= 226 ∙ 23/2
= 2599
- Using the Gauss formula
Sn = (x1 + xn ) ∙ n/2
yields for the last term xn after substituting the known values 2599 = (14 + xn ) ∙ 23/2
14 + xn = 2 ∙ 2599/23
xn = 2 ∙ 2599/23 - 14
= 5198/23 - 14
= 226 - 14
= 212
More Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series. Lessons and Learning Resources
Sequences and Series Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 12.2 | Working with Arithmetic and Geometric Series. How to find the Sum of the First n-Terms of a Series. | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 12.2.1 | Series versus Sequences | | |
| 12.2.2 | The Gauss Method and Arithmetic Series | | |
| 12.2.3 | An Alternative Formula for the Calculation of Sn in Arithmetic Series | | |
| 12.2.4 | Geometric Series | | |
| 12.2.5 | The Combination of Sequences and Series | | |
| 12.2.6 | Combined Series | | |
| 12.2.7 | The Practical Applications of Number Series and Sequences | | |
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