How to evaluate a geometric series?
Understand the Problem
The question is asking for the method to evaluate a geometric series, which involves finding the sum of the terms in the series given the first term and the common ratio.
Answer
The formula for the sum of a geometric series is given by $$ S_n = a \frac{1 - r^n}{1 - r} $$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
Answer for screen readers
The sum of the geometric series is given by
$$ S_n = a \frac{1 - r^n}{1 - r} $$.
Steps to Solve
- Identify the first term and common ratio
For a geometric series, the first term is often denoted as $a$ and the common ratio as $r$. You need to know these two values to proceed.
- Determine the number of terms
Next, identify the total number of terms in the geometric series, which we will denote as $n$.
- Use the formula for the sum of a geometric series
The sum $S_n$ of the first $n$ terms of a geometric series can be computed using the formula:
$$ S_n = a \frac{1 - r^n}{1 - r} $$
if $r \neq 1$.
- Substitute values into the formula
Replace $a$, $r$, and $n$ in the formula with the values you have identified to find the sum of the series.
- Calculate the final result
Perform the calculation based on your substituted values to obtain the sum of the geometric series.
The sum of the geometric series is given by
$$ S_n = a \frac{1 - r^n}{1 - r} $$.
More Information
A geometric series is vital in various fields, such as finance (for calculating compound interest), computer science (analysis of algorithms), and physics (wave functions). Understanding how to sum this series can be incredibly useful in real-world applications.
Tips
- Forgetting to check if the common ratio $r$ is equal to 1. If $r = 1$, the series is not geometric, and the formula must differ.
- Incorrectly substituting values for $a$, $r$, or $n$. It's essential to double-check that these values are accurate before calculating the sum.
