Math Problem Statement
Solution
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):
- Linear Sequence (LP): A sequence with a constant difference between consecutive terms.
- Quadratic Sequence (QP): A sequence where the second differences (the differences of the differences) are constant.
- 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?
Here are 5 questions to expand the topic:
- What is the general form of a linear sequence, and how is it derived?
- How can we identify a quadratic sequence just from its terms?
- What defines a geometric progression and how does it differ from an arithmetic sequence?
- Can a sequence be both linear and quadratic? Why or why not?
- 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.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Geometric Sequences
Quadratic Sequences
Formulas
Linear Sequence: a_n = a_1 + (n - 1) * d
Quadratic Sequence: a_n = an^2 + bn + c
Theorems
Arithmetic sequences involve constant first differences.
Quadratic sequences involve constant second differences.
Suitable Grade Level
Grades 7-9
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