https://file.aiquickdraw.com/mathbot/file/1736822626733r9uk75ke.jpg

Let's solve the problem step by step.

---

**Task:**  
For each sequence:
1. Determine the **common difference** (sequence generator).  
2. Write the formula for the **n-th term** in the sequence.

---

### Solution:

#### **(a) 4, 7, 10, 13, ...**
1. **Common difference**:  
   \( 7 - 4 = 3 \), \( 10 - 7 = 3 \), \( 13 - 10 = 3 \).  
   The common difference \( d = 3 \).

2. **Formula for the n-th term**:  
   The formula for the n-th term of an arithmetic sequence is:  
   \[
   t(n) = t(1) + (n-1)d
   \]  
   Substituting \( t(1) = 4 \) and \( d = 3 \):  
   \[
   t(n) = 4 + (n-1)(3) = 4 + 3n - 3 = 3n + 1
   \]

---

#### **(b) 3, 8, 13, ...**
1. **Common difference**:  
   \( 8 - 3 = 5 \), \( 13 - 8 = 5 \).  
   The common difference \( d = 5 \).

2. **Formula for the n-th term**:  
   Substituting \( t(1) = 3 \) and \( d = 5 \):  
   \[
   t(n) = 3 + (n-1)(5) = 3 + 5n - 5 = 5n - 2
   \]

---

#### **(c) 24, 19, 14, ...**
1. **Common difference**:  
   \( 19 - 24 = -5 \), \( 14 - 19 = -5 \).  
   The common difference \( d = -5 \).

2. **Formula for the n-th term**:  
   Substituting \( t(1) = 24 \) and \( d = -5 \):  
   \[
   t(n) = 24 + (n-1)(-5) = 24 - 5n + 5 = -5n + 29
   \]

---

#### **(d) 7, 9.5, 12, ...**
1. **Common difference**:  
   \( 9.5 - 7 = 2.5 \), \( 12 - 9.5 = 2.5 \).  
   The common difference \( d = 2.5 \).

2. **Formula for the n-th term**:  
   Substituting \( t(1) = 7 \) and \( d = 2.5 \):  
   \[
   t(n) = 7 + (n-1)(2.5) = 7 + 2.5n - 2.5 = 2.5n + 4.5
   \]

---

### Final Answers:

(a) \( t(n) = 3n + 1 \)  
(b) \( t(n) = 5n - 2 \)  
(c) \( t(n) = -5n + 29 \)  
(d) \( t(n) = 2.5n + 4.5 \)

---

Would you like more details on any part of this solution?  
Here are 5 related questions to explore further:  
1. How do you derive the general formula for an arithmetic sequence?  
2. What are other ways to find the common difference?  
3. Can the common difference be a fraction or decimal?  
4. How do you verify if a sequence is arithmetic?  
5. What happens if the common difference is 0?  

**Tip:** Always double-check the common difference before creating the formula!

Determine the common difference (sequence generator) for each arithmetic sequence listed below. Then write an equation for the n-th term in each sequence, keeping in mind that the first term of each sequence is t(1).

Explore how to find the common difference and write the n-th term formula for arithmetic sequences. This guide covers problems like 4, 7, 10, 13, and 3, 8, 13. Learn key concepts and solve problems step-by-step, perfect for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation