Directions: Answer the following and show your complete solution. 1 point for correct answer and solutions.
Arithmetic Sequence – is a sequence of terms that
have a common ___________ between them.
General Term:
an =
Geoemetric Sequence – is a sequence of terms that
have a common ___________ between them.
General Term:
an =
1. Are the following sequences arithmetic, geometric, or neither? If they are arithmetic, state the value of d. If
they are geometric, state r.
a. 2. 6, 12, 18, 24, ... d. 160, 80, 40, 20, ...
b. 6, 11, 17, ... e. -40, -25, -10, 5, ....
c. 2, 14, 98, 686, ... f. 7, -21, 63, -189,

### Arithmetic Sequence
An **arithmetic sequence** is a sequence of terms that have a common **difference** between them.

**General Term**:
\[ a_n = a_1 + (n - 1) \cdot d \]
where:
- \( a_n \) is the \( n \)th term,
- \( a_1 \) is the first term,
- \( d \) is the common difference.

### Geometric Sequence
A **geometric sequence** is a sequence of terms that have a common **ratio** between them.

**General Term**:
\[ a_n = a_1 \cdot r^{(n - 1)} \]
where:
- \( a_n \) is the \( n \)th term,
- \( a_1 \) is the first term,
- \( r \) is the common ratio.

### Problem 1: Sequence Classification and Identification of \( d \) or \( r \)

Let's determine if the given sequences are arithmetic, geometric, or neither.

#### a. Sequence: \( 6, 12, 18, 24, \dots \)

- **First Term**: \( a_1 = 6 \)
- **Common Difference**: \( d = 12 - 6 = 6 \)
  
This sequence has a common difference of 6, so it is an **arithmetic sequence** with \( d = 6 \).

#### b. Sequence: \( 6, 11, 17, \dots \)

- **First Term**: \( a_1 = 6 \)
- **Common Difference**: \( d = 11 - 6 = 5 \)

This sequence has a common difference of 5, so it is an **arithmetic sequence** with \( d = 5 \).

#### c. Sequence: \( 2, 14, 98, 686, \dots \)

- **First Term**: \( a_1 = 2 \)
- **Common Ratio**: \( r = \frac{14}{2} = 7 \)

This sequence has a common ratio of 7, so it is a **geometric sequence** with \( r = 7 \).

#### d. Sequence: \( 160, 80, 40, 20, \dots \)

- **First Term**: \( a_1 = 160 \)
- **Common Ratio**: \( r = \frac{80}{160} = \frac{1}{2} \)

This sequence has a common ratio of \( \frac{1}{2} \), so it is a **geometric sequence** with \( r = \frac{1}{2} \).

#### e. Sequence: \( -40, -25, -10, 5, \dots \)

- **First Term**: \( a_1 = -40 \)
- **Common Difference**: \( d = -25 - (-40) = 15 \)

This sequence has a common difference of 15, so it is an **arithmetic sequence** with \( d = 15 \).

#### f. Sequence: \( 7, -21, 63, -189, \dots \)

- **First Term**: \( a_1 = 7 \)
- **Common Ratio**: \( r = \frac{-21}{7} = -3 \)

This sequence has a common ratio of \( -3 \), so it is a **geometric sequence** with \( r = -3 \).

### Summary

- **a.** Arithmetic, \( d = 6 \)
- **b.** Arithmetic, \( d = 5 \)
- **c.** Geometric, \( r = 7 \)
- **d.** Geometric, \( r = \frac{1}{2} \)
- **e.** Arithmetic, \( d = 15 \)
- **f.** Geometric, \( r = -3 \)

Do you need more details or have any questions?

---

1. What is the general term formula for an arithmetic sequence?
2. How do you find the common ratio in a geometric sequence?
3. What is the next term in the sequence \( 6, 12, 18, 24, \dots \)?
4. Identify the type of sequence and find the 5th term of the sequence \( 2, 14, 98, 686, \dots \).
5. Can a sequence be both arithmetic and geometric? Explain why or why not.

**Tip**: When determining if a sequence is arithmetic, subtract consecutive terms to check for a constant difference. For geometric sequences, divide consecutive terms to check for a constant ratio.

Learn the definitions of arithmetic and geometric sequences with examples. Understand how to identify these sequences and find their common differences or ratios. Suitable for students in Grades 7-9.

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