To solve the sequence \(7, 12, 17, 22\), we need to determine the pattern or rule that defines the sequence. Let's examine the differences between consecutive terms:

- \(12 - 7 = 5\)
- \(17 - 12 = 5\)
- \(22 - 17 = 5\)

We see that each term increases by 5. Therefore, this is an arithmetic sequence with a common difference \(d = 5\).

To find the \(n\)-th term of an arithmetic sequence, we use the formula:

\[ a_n = a_1 + (n-1)d \]

Where:
- \(a_n\) is the \(n\)-th term
- \(a_1\) is the first term (7 in this case)
- \(d\) is the common difference (5 in this case)
- \(n\) is the term number

**1. Find the 5th term:**
\[ a_5 = 7 + (5-1) \cdot 5 = 7 + 4 \cdot 5 = 7 + 20 = 27 \]

**2. Find the 10th term:**
\[ a_{10} = 7 + (10-1) \cdot 5 = 7 + 9 \cdot 5 = 7 + 45 = 52 \]

**3. General formula for the \(n\)-th term:**
\[ a_n = 7 + (n-1) \cdot 5 \]

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the common difference in the sequence 3, 8, 13, 18?
2. How do you find the sum of the first \(n\) terms of an arithmetic sequence?
3. What is the general formula for the \(n\)-th term of a geometric sequence?
4. How would you identify if a sequence is arithmetic or geometric?
5. How do you find the common ratio in a geometric sequence?
6. Can you derive the formula for the sum of an arithmetic series?
7. How does the graph of an arithmetic sequence look?
8. What is the 20th term of the sequence 2, 6, 10, 14?

**Tip:** Always verify the pattern in a sequence by checking differences or ratios between multiple terms to ensure consistency.

Learn how to solve the sequence 7, 12, 17, 22 with a detailed step-by-step solution. This problem covers arithmetic sequences, including finding nth terms and identifying common differences. Suitable for students in Grades 7-9.

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