https://file.aiquickdraw.com/mathbot/file/1727110565766o2au1gps.jpg

Let’s break down and solve each of the questions visible in the image:

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### 1. **Conjecture about the next item in each sequence**

- (a) **Sequence**: \( 4, -2, 2, -8 \)
  - **Pattern**: The terms alternate between positive and negative, and the absolute value of the terms follows a pattern: \( 4, 2, 8 \) (dividing and multiplying by 2 alternately).
  - **Next term**: \( 2 \div 2 = 1 \), then apply the alternating sign: \( \mathbf{-1} \).

- (b) **Sequence**: \( 6, 11, \frac{5}{2}, 5, \frac{5}{4} \)
  - **Pattern**: Each term seems to be halving, except for a step between \( 11 \) and \( \frac{5}{2} \). It alternates between whole and fractional numbers.
  - **Next term**: The next fractional step would be \( \mathbf{5} \div 2 = 2.5 = \frac{5}{2} \).

- (c) **Sequence**: \( -2, -4, -8, -16 \)
  - **Pattern**: Each term is multiplied by 2.
  - **Next term**: \( -16 \times 2 = \mathbf{-32} \).

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### 2. **Counterexample to Rachel's conjecture**

- **Rachel's Conjecture**: Rachel thinks she is allergic to pear trees because she sneezes when they bloom.
- **Counterexample**: Rachel might be allergic to something else that blooms at the same time as pear trees, such as **grass** or **other plants** in the environment, indicating that the pear trees may not be the direct cause of her allergy.

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### 3. **Clark's statement about a parallelogram**

- **Clark's Statement**: A parallelogram is a quadrilateral with equal opposite angles.
- **If-Then Form**: If a quadrilateral has equal opposite angles, then it is a parallelogram.

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### 4. **Converse, Inverse, and Contrapositive of the Conditional Statement**

#### Conditional Statement:
**Original Statement**: "If you are seated in rows 10 to 20, you may now board."

- **Converse**: If you may now board, then you are seated in rows 10 to 20.
- **Inverse**: If you are not seated in rows 10 to 20, then you may not board.
- **Contrapositive**: If you may not board, then you are not seated in rows 10 to 20.

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### 5. **Converse, Inverse, and Contrapositive of Linda’s Medicine Warning**

#### Conditional Statement:
**Original Statement**: "If you will be driving, then you should not take this medicine."

- **Converse**: If you should not take this medicine, then you will be driving.
- **Inverse**: If you will not be driving, then you should take this medicine.
- **Contrapositive**: If you should take this medicine, then you will not be driving.

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Would you like further details on any specific part of the solution? Here are 5 related questions for more practice:

1. What is the next number in the sequence: \( 1, 4, 9, 16, \dots \)?
2. What is a different counterexample for Rachel’s conjecture about allergies?
3. How would you write the converse of the statement "If it rains, the ground gets wet"?
4. What is the inverse of the statement "If a figure is a square, then it has four right angles"?
5. Can you write the contrapositive of "If you study, you will pass the exam"?

**Tip**: When identifying patterns in sequences, look for arithmetic or geometric relationships, or alternating operations.

Solve the sequence patterns, provide a counterexample for Rachel's conjecture about pear trees, write the converse, inverse, and contrapositive of conditional statements, and more as seen in the image.

Understand how to solve sequence patterns, refute conjectures with counterexamples, and write the converse, inverse, and contrapositive of conditional statements. A comprehensive guide for high school students in Grades 9-11.

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