Sure, here is a sample work on linear and exponential sequences, which could be helpful for Algebra 1 students to understand these concepts better.
Sample Work: Linear and Exponential Sequences
Objective: Understand and apply the concepts of linear and exponential sequences.
I. Definitions
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Linear Sequence: A linear sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
General formula:
\( a_n = a_1 + (n - 1)d \)
where:- \( a_n \) = nth term
- \( a_1 \) = first term
- \( d \) = common difference
- \( n \) = term number
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Exponential Sequence: An exponential sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
General formula:
\( a_n = a_1 \cdot r^{(n-1)} \)
where:- \( a_n \) = nth term
- \( a_1 \) = first term
- \( r \) = common ratio
- \( n \) = term number
II. Examples
A. Linear Sequence Example
Problem: Find the first five terms of a linear sequence where the first term is \( 3 \) and the common difference is \( 2 \).
Solution:
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Start with the first term:
\( a_1 = 3 \) -
Calculate the next terms using the common difference \( d = 2 \):
- \( a_2 = a_1 + d = 3 + 2 = 5 \)
- \( a_3 = a_2 + d = 5 + 2 = 7 \)
- \( a_4 = a_3 + d = 7 + 2 = 9 \)
- \( a_5 = a_4 + d = 9 + 2 = 11 \)
First five terms:
\[ 3, 5, 7, 9, 11 \]
B. Exponential Sequence Example
Problem: Find the first five terms of an exponential sequence where the first term is \( 2 \) and the common ratio is \( 3 \).
Solution:
-
Start with the first term:
\( a_1 = 2 \) -
Calculate the next terms using the common ratio \( r = 3 \):
- \( a_2 = a_1 \cdot r = 2 \cdot 3 = 6 \)
- \( a_3 = a_2 \cdot r = 6 \cdot 3 = 18 \)
- \( a_4 = a_3 \cdot r = 18 \cdot 3 = 54 \)
- \( a_5 = a_4 \cdot r = 54 \cdot 3 = 162 \)
First five terms:
\[ 2, 6, 18, 54, 162 \]
III. Practice Problems
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Linear Sequence:
- Find the first five terms of a linear sequence with \( a_1 = 4 \) and \( d = 5 \).
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Exponential Sequence:
- Find the first five terms of an exponential sequence with \( a_1 = 1 \) and \( r = 2 \).
IV. Answer Key
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Linear Sequence Answer:
- \( 4, 9, 14, 19, 24 \)
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Exponential Sequence Answer:
- \( 1, 2, 4, 8, 16 \)
Conclusion
Understanding linear and exponential sequences is fundamental in algebra. By practicing these concepts, students can develop a solid foundation for more advanced mathematical topics. Make sure to try the practice problems and verify your answers!
This sample work includes definitions, examples, and practice problems that can be used in a classroom setting to teach students about linear and exponential sequences.