https://file.aiquickdraw.com/mathbot/file/17273478476705v9qhll9.jpg

Let's go over the problems in this image, which are focused on evaluating infinite geometric series.

### Formula for the sum of an infinite geometric series:
For an infinite geometric series with the first term \( a_1 \) and common ratio \( r \), the sum \( S \) is given by:
\[
S = \frac{a_1}{1 - r}, \quad \text{if} \ |r| < 1
\]
If \( |r| \geq 1 \), the series does not converge, so no sum can be found.

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### Solutions to the problems:

1. **\( 81 - 27 + 9 - 3 + \dots \)**:
   - First term \( a_1 = 81 \)
   - Common ratio \( r = \frac{-27}{81} = -\frac{1}{3} \)
   \[
   S = \frac{81}{1 - (-\frac{1}{3})} = \frac{81}{1 + \frac{1}{3}} = \frac{81}{\frac{4}{3}} = 81 \times \frac{3}{4} = 60.75
   \]

2. **\( 1 + 0.5 + 0.25 + 0.125 + \dots \)**:
   - First term \( a_1 = 1 \)
   - Common ratio \( r = \frac{0.5}{1} = 0.5 \)
   \[
   S = \frac{1}{1 - 0.5} = \frac{1}{0.5} = 2
   \]

3. **\( 3 + 6 + 12 + 24 + \dots \)**:
   - First term \( a_1 = 3 \)
   - Common ratio \( r = \frac{6}{3} = 2 \)
   - Since \( r = 2 \), the series diverges because \( |r| \geq 1 \). No sum exists.

4. **\( a_1 = -1, r = 3 \)**:
   - The series diverges because \( |r| \geq 1 \). No sum exists.

5. **\( a_1 = -3, r = 4 \)**:
   - The series diverges because \( |r| \geq 1 \). No sum exists.

6. **\( a_1 = 4, r = -\frac{3}{4} \)**:
   - First term \( a_1 = 4 \)
   - Common ratio \( r = -\frac{3}{4} \)
   \[
   S = \frac{4}{1 - (-\frac{3}{4})} = \frac{4}{1 + \frac{3}{4}} = \frac{4}{\frac{7}{4}} = 4 \times \frac{4}{7} = \frac{16}{7}
   \]

7. **\( a_1 = 5.5, r = 0.5 \)**:
   - First term \( a_1 = 5.5 \)
   - Common ratio \( r = 0.5 \)
   \[
   S = \frac{5.5}{1 - 0.5} = \frac{5.5}{0.5} = 11
   \]

8. **Summation \( \sum_{i=1}^{\infty} \left( \frac{1}{3} \right)^{i-1} \)**:
   - This is a geometric series with \( a_1 = 1 \) (first term is \( \left(\frac{1}{3}\right)^0 = 1 \)) and \( r = \frac{1}{3} \)
   \[
   S = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}
   \]

9. **Summation \( \sum_{i=1}^{\infty} 2^{i-1} \)**:
   - This is a geometric series with \( a_1 = 1 \) (first term is \( 2^0 = 1 \)) and \( r = 2 \)
   - Since \( r = 2 \), the series diverges because \( |r| \geq 1 \). No sum exists.

10. **Summation \( \sum_{i=1}^{\infty} 4^{i-1} \)**:
    - This is a geometric series with \( a_1 = 1 \) (first term is \( 4^0 = 1 \)) and \( r = 4 \)
    - Since \( r = 4 \), the series diverges because \( |r| \geq 1 \). No sum exists.

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Would you like further details or clarifications?

### 5 Related Questions:
1. What conditions are needed for a geometric series to converge?
2. How does the common ratio affect the sum of an infinite series?
3. Can geometric series with a common ratio greater than 1 ever have a finite sum?
4. How can you express a recurring decimal as an infinite geometric series?
5. What is the significance of geometric series in real-life applications?

### Tip:
When the common ratio \( |r| \) is less than 1, the series converges to a finite value, but when \( |r| \geq 1 \), the series diverges (no finite sum).

Activity 2. Sum to Infinity! Direction: Evaluate each infinite geometric series described.

Learn how to evaluate infinite geometric series using the formula for series sum. This activity covers convergence and divergence of geometric series, providing detailed solutions for sums with various common ratios. Ideal for high school students in Grades 10-12.

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