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:

$S_n = a + ar + ar^2 + ar^3 + \dots + ar^{n-1}$

Where:

• $a$ is the first term,
• $r$ is the common ratio (the factor between consecutive terms),
• $n$ is the number of terms.

The formula for the sum of the first $n$ terms of a geometric series is:

$S_n = \frac{a(1 - r^n)}{1 - r}$

Important conditions:

• If $r = 1$, the formula simplifies to $S_n = n \cdot a$, since each term in the series is the same.
• This formula works when $r \neq 1$.

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 ($|r| < 1$).

The general form of the infinite geometric series is:

$S_\infty = a + ar + ar^2 + ar^3 + \dots$

The formula for the sum of an infinite geometric series is:

$S_\infty = \frac{a}{1 - r} \quad \text{for} \quad |r| < 1$

Example Problems

Example 1: Finite Geometric Series

Find the sum of the first 5 terms of the series: $2 + 6 + 18 + 54 + 162$.

Here:

• $a = 2$ (first term),
• $r = 3$ (common ratio),
• $n = 5$ (number of terms).

Using the formula for the sum of a finite geometric series:

$S_5 = \frac{2(1 - 3^5)}{1 - 3} = \frac{2(1 - 243)}{-2} = \frac{2(-242)}{-2} = 242$

So, the sum of the first 5 terms is 242.

Example 2: Infinite Geometric Series

Find the sum of the infinite series: $4 + 2 + 1 + \frac{1}{2} + \dots$.

Here:

• $a = 4$ (first term),
• $r = \frac{1}{2}$ (common ratio).

Since $|r| < 1$, we can use the infinite series formula:

$S_\infty = \frac{4}{1 - \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 8$

So, the sum of the infinite series is 8.

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Would you like more examples or need further clarification? Here are some related questions you can explore:

1. How does the sum of a geometric series change if the common ratio is negative?
2. How can you determine if a geometric series converges or diverges?
3. What is the sum of an infinite geometric series when the common ratio is greater than 1?
4. How do you apply the sum formula to series with fractional terms?
5. 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 $|r| \geq 1$, the series does not have a finite sum.