https://file.aiquickdraw.com/mathbot/file/1726248890834jxj4plsu.jpg

Let's analyze each of the sequences from the image to determine if they are **Linear (LP)**, **Quadratic (QP)**, or **Neither Linear nor Quadratic (NLQP)**:

1. **Linear Sequence (LP)**: A sequence with a constant difference between consecutive terms.
2. **Quadratic Sequence (QP)**: A sequence where the second differences (the differences of the differences) are constant.
3. **Neither Linear nor Quadratic (NLQP)**: If neither of the above properties apply.

### (a) Sequence: 1, 1, 2, 3, 5, 8, ...
- First differences: 1-1=0, 2-1=1, 3-2=1, 5-3=2, 8-5=3
- The first differences are not constant, and the second differences (1-0=1, 1-1=0, 2-1=1, 3-2=1) are also not constant.
- This is the **Fibonacci Sequence**.
- **Answer: NLQP**

### (b) Sequence: 4, 7, 10, 13, ...
- First differences: 7-4=3, 10-7=3, 13-10=3
- The first differences are constant (3).
- **Answer: LP**

### (c) Sequence: 2, 4, 8, 16, ...
- First differences: 4-2=2, 8-4=4, 16-8=8
- Second differences: 4-2=2, 8-4=4
- The second differences are not constant, but the ratio between consecutive terms is constant (geometric progression with ratio 2).
- **Answer: NLQP**

### (d) Sequence: 10, 8, 6, 4, 2, ...
- First differences: 8-10=-2, 6-8=-2, 4-6=-2, 2-4=-2
- The first differences are constant (-2).
- **Answer: LP**

### (e) Sequence: 1, 3, 6, 10, 15, ...
- First differences: 3-1=2, 6-3=3, 10-6=4, 15-10=5
- Second differences: 3-2=1, 4-3=1, 5-4=1
- The second differences are constant (1), so this is a quadratic sequence.
- **Answer: QP**

### (f) Sequence: 160, 80, 40, 20, ...
- First differences: 80-160=-80, 40-80=-40, 20-40=-20
- The first differences are not constant, but the ratio between consecutive terms is 1/2 (geometric sequence).
- **Answer: NLQP**

### (g) Sequence: 2, 5, 10, 17, ...
- First differences: 5-2=3, 10-5=5, 17-10=7
- Second differences: 5-3=2, 7-5=2
- The second differences are constant (2), so this is a quadratic sequence.
- **Answer: QP**

### (h) Sequence: 1, 3, 5, 7, 9, ...
- First differences: 3-1=2, 5-3=2, 7-5=2, 9-7=2
- The first differences are constant (2).
- **Answer: LP**

### Final Answers:
- (a) NLQP
- (b) LP
- (c) NLQP
- (d) LP
- (e) QP
- (f) NLQP
- (g) QP
- (h) LP

Would you like more details on any specific sequence or concept?

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Here are 5 questions to expand the topic:
1. What is the general form of a linear sequence, and how is it derived?
2. How can we identify a quadratic sequence just from its terms?
3. What defines a geometric progression and how does it differ from an arithmetic sequence?
4. Can a sequence be both linear and quadratic? Why or why not?
5. How does the Fibonacci sequence grow, and why is it categorized as NLQP?

**Tip**: For quadratic sequences, always check the second differences to confirm their nature.

For each of the sequences given, decide whether it is linear or quadratic. Type LP for linear, QP for Quadratic, and NLQP if it is not linear nor quadratic.

Analyze various sequences to determine if they are linear, quadratic, or neither. This problem involves key mathematical concepts such as arithmetic and geometric sequences, as well as quadratic progression. Suitable for students in Grades 7-9.

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