The formula for the sum of a **geometric series** depends on whether the series is **finite** or **infinite**. Here are the formulas:

### 1. **Finite Geometric Series Formula:**
For a geometric series with a first term \( a \), common ratio \( r \), and \( n \) terms, the sum \( S_n \) is given by:

\[
S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{if} \quad r \neq 1
\]

Where:
- \( S_n \) is the sum of the first \( n \) terms.
- \( a \) is the first term.
- \( r \) is the common ratio (i.e., the factor by which each term is multiplied to get the next term).
- \( n \) is the number of terms.

### 2. **Infinite Geometric Series Formula:**
For an **infinite geometric series** where \( |r| < 1 \) (i.e., the absolute value of the common ratio is less than 1), the sum \( S_{\infty} \) is:

\[
S_{\infty} = \frac{a}{1 - r} \quad \text{if} \quad |r| < 1
\]

Where:
- \( S_{\infty} \) is the sum of the infinite series.
- \( a \) is the first term.
- \( r \) is the common ratio.

#### Example:
For the geometric series \( 3 + 6 + 12 + 24 + \dots \), the first term \( a = 3 \), and the common ratio \( r = 2 \).

- For a **finite sum of 5 terms**:
\[
S_5 = 3 \cdot \frac{1 - 2^5}{1 - 2} = 3 \cdot \frac{1 - 32}{-1} = 3 \cdot 31 = 93
\]

- For an **infinite series** (assuming \( |r| < 1 \), but here it's not applicable since \( r = 2 \) is greater than 1).

### Would you like to see how this formula applies to a specific problem?

### 5 Relative Questions:
1. What is the relationship between the common ratio \( r \) and the behavior of a geometric series?
2. How do you find the first term and common ratio in a given geometric sequence?
3. Can a geometric series have negative or fractional terms?
4. How do geometric series formulas change when working with infinite series?
5. What are the real-world applications of geometric series in finance or physics?

### Tip:
For an infinite geometric series, ensure the common ratio \( r \) is between -1 and 1, or else the series will not converge (i.e., it won't have a finite sum).

Explore the formula for the sum of a geometric series and its applications. Learn how to calculate both finite and infinite geometric series sums with detailed examples and explanations. Suitable for high school students and anyone interested in understanding geometric series.

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