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Welcome to our Math lesson on Geometric Series, this is the fourth 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.
Geometric Series Explained
A geometric series Sn represents the sum of the first n terms of a geometric sequence (progression). For example, if we have the geometric sequence with five terms
1,2,4,8,16
then, the sum of these terms represents the corresponding series Sn, where n = 5. Thus,
Sn = 1 + 2 + 4 + 8 + 16
= 31
Let's explain how to find the sum of the first n terms in a geometric series. Thus,
S1 = x1
S2 = x1 + x2 = x1 + R ∙ x1
S3 = x1 + x2 + x3 = x1 + R ∙ x1 + R2 ∙ x1
⋮
Sn = x1 + x2 + x3 + ⋯ + xn = x1 + R ∙ x1 + R2 ∙ x1 + ⋯ + Rn - 1 ∙ x1
Multiplying the last expression by the common ratio R yields
R ∙ Sn = R ∙ (x1 + x2 + x3 + ⋯ + xn ) = R ∙ x1 + R2 ∙ x1 + R3 ∙ x1 + ⋯ + Rn ∙ x1
Subtracting the previous expression from the last one yields
R ∙ Sn - Sn = (Rx1 + R2 x1 + R3 x1 + ⋯ + Rn x1 ) - (x1 + Rx1 + R2 x1 + ⋯ + Rn - 1 x1)
= Rn x1 - x1
Thus,
Sn (R - 1) = x1 (Rn - 1)
In this way, we obtain the general formula for calculating the sum of the first n terms of a geometric series
Sn = x1 (Rn - 1)/ R - 1
Multiplying the above formula up and down by -1 gives another version of the sum of the first n terms of a geometric progression (series), i.e.
Sn = x1 (1 - Rn)/1 - R
This version is used in decreasing geometric sequences, where the common ratio R is smaller than 1.
Example 5
Calculate the sum of the first 25 terms of the geometric series
2, 6, 18, 54, …
Solution 5
We have x1 = 2, R = 3 and n = 25. Thus, applying the first formula for calculation of the first n terms of a geometric series, yields
Sn = x1 (Rn - 1)/R - 1
S25 = 2 ∙ (325- 1)/3 - 1
= 325 - 1
= 847,288,609,443 - 1
= 847,288,609,442
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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