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 $\frac{1}{16}, \frac{1}{32}, \frac{1}{64}$.

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: $5, -1, 2$.

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 $10, 20, 40$.

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.