Math Problem Statement
how to find sum of geometric series
Solution
To find the sum of a geometric series, you can use a specific formula depending on whether the series is finite or infinite. Here's how to approach both:
1. Sum of a Finite Geometric Series
A finite geometric series is a series that has a fixed number of terms. The general form of a geometric series is:
Where:
- is the first term,
- is the common ratio (the factor between consecutive terms),
- is the number of terms.
The formula for the sum of the first terms of a geometric series is:
Important conditions:
- If , the formula simplifies to , since each term in the series is the same.
- This formula works when .
2. Sum of an Infinite Geometric Series
An infinite geometric series has an infinite number of terms, and it converges (i.e., has a finite sum) only when the absolute value of the common ratio is less than 1 ().
The general form of the infinite geometric series is:
The formula for the sum of an infinite geometric series is:
Example Problems
Example 1: Finite Geometric Series
Find the sum of the first 5 terms of the series: .
Here:
- (first term),
- (common ratio),
- (number of terms).
Using the formula for the sum of a finite geometric series:
So, the sum of the first 5 terms is 242.
Example 2: Infinite Geometric Series
Find the sum of the infinite series: .
Here:
- (first term),
- (common ratio).
Since , we can use the infinite series formula:
So, the sum of the infinite series is 8.
Would you like more examples or need further clarification? Here are some related questions you can explore:
- How does the sum of a geometric series change if the common ratio is negative?
- How can you determine if a geometric series converges or diverges?
- What is the sum of an infinite geometric series when the common ratio is greater than 1?
- How do you apply the sum formula to series with fractional terms?
- How can the geometric series formula be used in real-world problems, such as calculating compound interest?
Tip: When solving geometric series problems, always check the common ratio, especially for infinite series. If , the series does not have a finite sum.
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Math Problem Analysis
Mathematical Concepts
Algebra
Geometric Series
Sequences and Series
Formulas
Sum of a finite geometric series: S_n = a(1 - r^n) / (1 - r) for r ≠ 1
Sum of an infinite geometric series: S_∞ = a / (1 - r) for |r| < 1
Theorems
Convergence of Infinite Geometric Series (for |r| < 1)
Suitable Grade Level
Grades 9-12