https://file.aiquickdraw.com/mathbot/file/17265005858111r8887mt.jpg

Let's analyze and complete the table in the image step by step:

### Sequences in the Table:
- **Sequence A**: \( 6, 10, 14, 18, \dots \) (given)
- **Sequence B**: \( 7, 11, 15, 19, \dots \) (given)
- **Sequence C**: Not provided (we will find it).
- **Sequence D**: Not provided (we will find it).
- **Sequence E**: \( -6, -10, -14, -18, \dots \) (given)

### Relationships and Rules:

1. **Sequence A** has a term-to-term rule of **adding 4** between consecutive terms: 
   - \( 6 + 4 = 10 \), 
   - \( 10 + 4 = 14 \),
   - \( 14 + 4 = 18 \), 
   and so on.

2. **Sequence B** is related to Sequence A by adding **1 to each term in Sequence A**:
   - \( 6 + 1 = 7 \), 
   - \( 10 + 1 = 11 \), 
   - \( 14 + 1 = 15 \), 
   - \( 18 + 1 = 19 \).

   Therefore, Sequence B follows the same term-to-term rule as Sequence A, i.e., **add 4**.

3. **Sequence C** is described as **adding 4** to each term in Sequence A:
   - \( 6 + 4 = 10 \),
   - \( 10 + 4 = 14 \),
   - \( 14 + 4 = 18 \),
   - \( 18 + 4 = 22 \).

   Hence, Sequence C: \( 10, 14, 18, 22, \dots \).

4. **Sequence D** is described as **doubling each term in Sequence A**:
   - \( 6 \times 2 = 12 \),
   - \( 10 \times 2 = 20 \),
   - \( 14 \times 2 = 28 \),
   - \( 18 \times 2 = 36 \).

   Thus, Sequence D: \( 12, 20, 28, 36, \dots \).

5. **Sequence E** is \( -6, -10, -14, -18, \dots \) and appears to follow a rule of **subtracting 4** (or adding negative 4) between consecutive terms:
   - \( -6 - 4 = -10 \),
   - \( -10 - 4 = -14 \),
   - \( -14 - 4 = -18 \).

---

### Completing the Table:
- **Sequence C**: \( 10, 14, 18, 22, \dots \)
- **Sequence D**: \( 12, 20, 28, 36, \dots \)
- **5th term of Sequence A**: \( 18 + 4 = 22 \)
- **5th term of Sequence B**: \( 19 + 4 = 23 \)
- **5th term of Sequence C**: \( 22 + 4 = 26 \)
- **5th term of Sequence D**: \( 36 + 8 = 44 \)
- **5th term of Sequence E**: \( -18 - 4 = -22 \)

---

Let me know if you'd like further explanation on any of the sequences or steps involved! Here are some related questions:

1. How can you derive the formula for the \(n\)-th term of an arithmetic sequence?
2. Can you write the explicit formula for Sequence B based on its term-to-term rule?
3. What is the difference between geometric and arithmetic sequences?
4. How do the properties of the sequences change when you double or halve terms?
5. Can sequences like these be used in modeling real-world problems?

**Tip**: Arithmetic sequences have a constant difference between consecutive terms, and this can be used to quickly generate any term in the sequence!

Complete the table by analyzing and filling in the missing values based on the provided number sequences and their relationships.

Learn how to complete a table for arithmetic sequences by understanding the relationships between the sequences. The problem involves adding, subtracting, and doubling terms, making it suitable for Grades 6-8 students learning arithmetic sequences and term-to-term rules.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation