https://file.aiquickdraw.com/mathbot/file/17257525707849jyrdzmr.jpg

Let's go through the questions in the image step by step.

### 1. **Find the specified sum for each of the following geometric series:**
   
   **a. \( 5 + 10 + 20 + \dots \) find \( S_{10} \)**

   This is a geometric series with:
   - First term \( a = 5 \)
   - Common ratio \( r = \frac{10}{5} = 2 \)

   The sum of the first \( n \) terms of a geometric series is given by the formula:
   \[
   S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{(for } r \neq 1)
   \]
   For \( S_{10} \), we plug in:
   - \( a = 5 \)
   - \( r = 2 \)
   - \( n = 10 \)
   \[
   S_{10} = \frac{5(1 - 2^{10})}{1 - 2}
   \]

   **b. \( 1 - 3 + 9 - \dots \) find \( S_6 \)**

   This is another geometric series with alternating signs:
   - First term \( a = 1 \)
   - Common ratio \( r = -3 \)

   Again, using the formula for the sum of the first \( n \) terms:
   \[
   S_6 = \frac{1(1 - (-3)^6)}{1 - (-3)}
   \]

   **c. \( \frac{4}{3} + \frac{2}{3} + \frac{1}{3} + \dots \) find \( S_5 \)**

   This geometric series has:
   - First term \( a = \frac{4}{3} \)
   - Common ratio \( r = \frac{1}{2} \)

   Applying the sum formula:
   \[
   S_5 = \frac{\frac{4}{3}(1 - \left(\frac{1}{2}\right)^5)}{1 - \frac{1}{2}}
   \]

### 2. **For the geometric sequence \( 2, 6, 18, \dots \) write an expression for:**

   **a. The \( n \)-th term**

   In a geometric sequence, the \( n \)-th term is given by:
   \[
   a_n = a \cdot r^{n-1}
   \]
   Here, \( a = 2 \) and \( r = \frac{6}{2} = 3 \). So the expression for the \( n \)-th term is:
   \[
   a_n = 2 \cdot 3^{n-1}
   \]

   **b. The sum of the first \( n \) terms**

   Using the sum formula again:
   \[
   S_n = \frac{a(1 - r^n)}{1 - r}
   \]
   For this series, \( a = 2 \) and \( r = 3 \), so:
   \[
   S_n = \frac{2(1 - 3^n)}{1 - 3}
   \]

### 3. **For the geometric sequence \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots \), write an expression for:**

   **a. The \( n \)-th term**

   Here:
   - First term \( a = 1 \)
   - Common ratio \( r = \frac{1}{2} \)

   So the \( n \)-th term is:
   \[
   a_n = 1 \cdot \left(\frac{1}{2}\right)^{n-1}
   \]
   Simplifying:
   \[
   a_n = \left(\frac{1}{2}\right)^{n-1}
   \]

   **b. The sum of the first \( n \) terms**

   Using the sum formula for geometric series:
   \[
   S_n = \frac{1(1 - \left(\frac{1}{2}\right)^n)}{1 - \frac{1}{2}}
   \]
   Simplifying:
   \[
   S_n = 2 \left(1 - \left(\frac{1}{2}\right)^n\right)
   \]

---

Would you like a detailed breakdown of any particular solution or calculation?

Here are 5 relative questions for further practice:
1. How would the sum formula change if the common ratio is negative?
2. Can you find \( S_5 \) for the sequence \( 1 + 2 + 4 + 8 + 16 + \dots \)?
3. For the series \( 2, 6, 18, 54, \dots \), calculate the sum of the first 7 terms.
4. How would you calculate the 12th term of the series \( 3 + 6 + 12 + 24 + \dots \)?
5. What happens to the sum of an infinite geometric series if \( |r| < 1 \)?

**Tip**: When working with geometric series, always check if the common ratio is between -1 and 1, as this allows for the sum of an infinite series to exist.

Find the specified sum for each of the following geometric series and write an expression for the given sequences.

Learn how to calculate the sum for specific geometric series and write expressions for geometric sequences. This guide includes formulas for the summation of geometric series and for the n-th term in a sequence, suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation