Question

02052 Algebra 1 A - Linear & Exponential Sequences Apply Make a Sample Work about this

Answer 1

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

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**Objective:** Understand and apply the concepts of linear and exponential sequences.

### I. Definitions

1. **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

2. **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:**
1. Start with the first term:
\( a_1 = 3 \)

2. 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:**
1. Start with the first term:
\( a_1 = 2 \)

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

1. **Linear Sequence:**
- Find the first five terms of a linear sequence with \( a_1 = 4 \) and \( d = 5 \).

2. **Exponential Sequence:**
- Find the first five terms of an exponential sequence with \( a_1 = 1 \) and \( r = 2 \).

### IV. Answer Key

1. **Linear Sequence Answer:**
- \( 4, 9, 14, 19, 24 \)

2. **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!

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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.