The sum of a limited geometric series is a mathematical operation used to find the total of a set of numbers that follow a geometric pattern. This series involves four key entities: the first term (a), the common ratio (r), the number of terms (n), and the sum (S). The common ratio represents the constant multiplier applied to each term to obtain the subsequent term. The number of terms determines the length of the series, and the sum represents the total value of all the terms combined.
The Sum of a Limited Geometric Series
A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value, called the common ratio. Geometric series are common in mathematics and have many applications in the physical and financial worlds.
The sum of a limited geometric series is given by the following formula:
(1) S = a * (1 – r^n) / (1 – r)
where
- S is the sum of the series
- a is the first term in the series
- r is the common ratio
- n is the number of terms in the series
For example, if we have a geometric series with a first term of 2 and a common ratio of 3, the sum of the first 5 terms is:
(2) S = 2 * (1 – 3^5) / (1 – 3) = 2 * (1 – 243) / (-2) = 484
Geometric series can also be represented in table form. The following table shows the first few terms of a geometric series with a first term of 2 and a common ratio of 3:
| Term | Value |
|---|---|
| 1 | 2 |
| 2 | 6 |
| 3 | 18 |
| 4 | 54 |
| 5 | 162 |
As you can see, each term in the series is found by multiplying the previous term by 3.
Geometric series are powerful tools that can be used to solve a variety of problems. The formula for the sum of a limited geometric series is essential for understanding how these series work and for using them to solve problems.
Question 1:
What is the formula for the sum of a limited geometric series?
Answer:
The sum of a limited geometric series with first term a, common ratio r, and n terms is given by the formula:
Subject: Sum of a geometric series
Predicate: Formula
Object: S = a(1 – r^n)/(1 – r)
Question 2:
How can I determine the convergence of a geometric series?
Answer:
A geometric series converges if the absolute value of the common ratio is less than 1, meaning that the terms decrease in size as n increases.
Subject: Convergence of a geometric series
Predicate: Condition
Object: |r| < 1
Question 3:
What is the relationship between the sum of a geometric series and its first term and common ratio?
Answer:
The sum of a geometric series is directly proportional to its first term and inversely proportional to its common ratio when the common ratio is less than 1.
Subject: Sum of a geometric series
Predicate: Relationship
Object: S = a(1 – r^n)/(1 – r), where |r| < 1
And that’s it, folks! We’ve covered the basics of finding the sum of a not-so-complicated geometric series. It might still sound intimidating, but trust me, it’s like learning to ride a bike – once you get the hang of it, you’ll feel like a pro. Thanks for hanging in there with me, and if you’re up for more mathematical adventures, be sure to drop by later. Until then, stay awesome and keep your geometer spirits high!