Numerical Patterns worksheet with questions on identifying sequence rules and completing tables.

Problem Analysis:

The worksheet involves identifying numerical patterns and completing tables based on given rules. Let's solve each question step by step.

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Question 1: What is the rule for the sequence? 2, 4, 8, 16...

- Observation: Each term is obtained by multiplying the previous term by 2.
- Rule: Start with 2, multiply by 2 to get the next term.
- Answer: Multiply by 2.

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Question 2: What is the rule for the sequence? 30, 34, 38, 42...

- Observation: Each term increases by 4.
- Rule: Start with 30, add 4 to get the next term.
- Answer: Add 4.

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Question 3: What is the rule for the sequence? 200, 100, 50, 25...

- Observation: Each term is obtained by dividing the previous term by 2.
- Rule: Start with 200, divide by 2 to get the next term.
- Answer: Divide by 2.

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Question 4: What is the rule for the sequence? 45, 40, 35, 30...

- Observation: Each term decreases by 5.
- Rule: Start with 45, subtract 5 to get the next term.
- Answer: Subtract 5.

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Question 5: What is the rule for the sequence? 18, 12, 6, 0...

- Observation: Each term decreases by 6.
- Rule: Start with 18, subtract 6 to get the next term.
- Answer: Subtract 6.

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Question 6: Complete the table

| Number of Weeks | Number of Cookies |
|-----------------|--------------------|
| 4 | 2 |
| 8 | 4 |
| 12 | 8 |

- Observation: The number of cookies doubles every 4 weeks.
- Rule: Number of cookies = \( \frac{\text{Number of Weeks}}{2} \).
- Completed Table:
| Number of Weeks | Number of Cookies |
|-----------------|--------------------|
| 4 | 2 |
| 8 | 4 |
| 12 | 8 |

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Question 7: Complete the table

| Number of Teachers | Number of Students |
|--------------------|--------------------|
| 100 | 3 |
| 75 | 2 |
| 4 | 3 |

- Observation: The relationship between teachers and students is not straightforward from the given data. However, we can infer a pattern if more context is provided. Since no clear rule is evident, we assume the values are fixed as given.
- Completed Table (as given):
| Number of Teachers | Number of Students |
|--------------------|--------------------|
| 100 | 3 |
| 75 | 2 |
| 4 | 3 |

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Question 8: Complete the table

| Number of Red Marbles | Number of Blue Marbles |
|-----------------------|------------------------|
| 80 | 60 |
| 60 | 40 |
| 40 | 80 |

- Observation: The total number of marbles remains constant (140). For every 4 red marbles, there are 3 blue marbles.
- Rule: Ratio of red to blue marbles = 4:3.
- Completed Table:
| Number of Red Marbles | Number of Blue Marbles |
|-----------------------|------------------------|
| 80 | 60 |
| 60 | 40 |
| 40 | 80 |

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Question 9: Write the first 4 numbers of a sequence with the rule: Start of 5, Increased by 6

- Rule: Start with 5, add 6 to get the next term.
- Sequence: 5, 11, 17, 23.
- Answer: 5, 11, 17, 23.

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Question 10: Write the first 4 numbers of a sequence with the rule: Start of 20, decrease by 7

- Rule: Start with 20, subtract 7 to get the next term.
- Sequence: 20, 13, 6, -1.
- Answer: 20, 13, 6, -1.

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Final Answers:

1. Multiply by 2.
2. Add 4.
3. Divide by 2.
4. Subtract 5.
5. Subtract 6.
6. Completed table:
| Number of Weeks | Number of Cookies |
|-----------------|--------------------|
| 4 | 2 |
| 8 | 4 |
| 12 | 8 |
7. Completed table (as given):
| Number of Teachers | Number of Students |
|--------------------|--------------------|
| 100 | 3 |
| 75 | 2 |
| 4 | 3 |
8. Completed table:
| Number of Red Marbles | Number of Blue Marbles |
|-----------------------|------------------------|
| 80 | 60 |
| 60 | 40 |
| 40 | 80 |
9. 5, 11, 17, 23.
10. 20, 13, 6, -1.

Boxed Final Answer:
\[
\boxed{
\begin{array}{l}
1. \text{Multiply by 2.} \\
2. \text{Add 4.} \\
3. \text{Divide by 2.} \\
4. \text{Subtract 5.} \\
5. \text{Subtract 6.} \\
6. \begin{array}{c|c}
\text{Number of Weeks} & \text{Number of Cookies} \\
\hline
4 & 2 \\
8 & 4 \\
12 & 8 \\
\end{array} \\
7. \begin{array}{c|c}
\text{Number of Teachers} & \text{Number of Students} \\
\hline
100 & 3 \\
75 & 2 \\
4 & 3 \\
\end{array} \\
8. \begin{array}{c|c}
\text{Number of Red Marbles} & \text{Number of Blue Marbles} \\
\hline
80 & 60 \\
60 & 40 \\
40 & 80 \\
\end{array} \\
9. 5, 11, 17, 23. \\
10. 20, 13, 6, -1.
\end{array}
}
\]