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I can see the image you've uploaded contains some sequences and their corresponding questions. Let me summarize the questions and help with the solutions. Based on what I can read:

### 2. A sequence starts with \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots \)
   - **a. Give a rule that the sequence follows.**
     - It looks like the rule is to multiply the previous term by \( \frac{1}{2} \) to get the next term.
   
   - **b. Follow your rule to write the next three terms.**
     - The next term after \( \frac{1}{8} \) is \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \).
     - Continuing, \( \frac{1}{16} \times \frac{1}{2} = \frac{1}{32} \).
     - Then, \( \frac{1}{32} \times \frac{1}{2} = \frac{1}{64} \).
     - So, the next three terms are \( \frac{1}{16}, \frac{1}{32}, \frac{1}{64} \).

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### 3. A sequence of numbers follows the rule: multiply the previous term by -2 and add 3.
   - **a. The fourth term is -7. Give the next three terms.**
     - To find the next term, we apply the rule.
     - If the fourth term is \(-7\):
       \[
       \text{Next term} = (-7 \times -2) + 3 = 14 + 3 = 17
       \]
     - Then, for the term after 17:
       \[
       \text{Next term} = (17 \times -2) + 3 = -34 + 3 = -31
       \]
     - Finally, for the next term:
       \[
       \text{Next term} = (-31 \times -2) + 3 = 62 + 3 = 65
       \]
     - So, the next three terms are: \( 17, -31, 65 \).

   - **b. Give the 3 terms that came before -7 in the sequence.**
     - To work backwards, we reverse the rule. If the rule is "multiply by -2 and add 3," to find previous terms, we subtract 3, then divide by -2.
     - The term before -7:
       \[
       \text{Previous term} = \frac{-7 - 3}{-2} = \frac{-10}{-2} = 5
       \]
     - The term before 5:
       \[
       \text{Previous term} = \frac{5 - 3}{-2} = \frac{2}{-2} = -1
       \]
     - The term before -1:
       \[
       \text{Previous term} = \frac{-1 - 3}{-2} = \frac{-4}{-2} = 2
       \]
     - So, the three terms before -7 are: \( 5, -1, 2 \).

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### 4. A sequence starts with \( 0, 5, \dots \)
   - **a. Give a rule and the next three terms for the sequence.**
     - The rule appears to be: "Add 5 to each term."
     - So, starting from \( 0, 5 \):
       - \( 5 + 5 = 10 \)
       - \( 10 + 5 = 15 \)
       - \( 15 + 5 = 20 \)
     - The next three terms are \( 10, 15, 20 \).

   - **b. Give a different rule and the next three terms for that rule.**
     - A different rule could be: "Multiply each term by 2."
     - Starting from \( 0, 5 \):
       - \( 5 \times 2 = 10 \)
       - \( 10 \times 2 = 20 \)
       - \( 20 \times 2 = 40 \)
     - The next three terms would be \( 10, 20, 40 \).

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If you'd like to go over any of the steps or concepts in more detail, feel free to ask!

### 5 Related Questions:
1. What other rules can you find to generate new sequences starting from \( 0 \) and \( 5 \)?
2. Can you create a sequence that alternates between positive and negative numbers?
3. How would you determine a rule if the sequence were decreasing instead of increasing?
4. What happens to the terms if you add both a multiplication and an addition step to the rule?
5. Can you identify any other real-world patterns that follow the rules of sequences?

### Tip:  
When working with sequences, always identify if the rule involves addition, multiplication, or both, and check if there is a constant pattern across the terms.

A sequence starts with 1/2, 1/4, 1/8, ..., a rule for this sequence and the next three terms; another sequence where the fourth term is -7, and find the next three and previous three terms; a sequence starting with 0, 5, ..., with a rule and two possible sets of next three terms based on different rules.

Explore sequence patterns starting with fractions and integers. This guide explains how to derive the rules for sequences, calculate the next three terms, and apply arithmetic and geometric progressions. Suitable for Grades 6-8.

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