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From the image, it looks like we are dealing with arithmetic sequence problems where we need to solve for a specific term, the common difference, or the number of terms in a sequence. Here's a breakdown for each problem shown:

**Problem 28:**
Given the first term \( a_1 = 5 \) and the 10th term \( a_{10} = 85 \), we need to find the common difference \( d \).

Using the formula for the nth term of an arithmetic sequence, \( a_n = a_1 + (n-1) \cdot d \), we can plug in the known values:
\[ a_{10} = 5 + (10-1) \cdot d = 85 \]

**Problem 29:**
Given the first term \( a_1 = 4 \) and the 12th term \( a_{12} = 26 \), we need to find the common difference \( d \) as well.

Using the same formula:
\[ a_{12} = 4 + (12-1) \cdot d = 26 \]

**Problem 30:**
Given the first term \( a_1 = -1 \), the common difference \( d = 5 \), and the nth term \( a_n = 34 \), we need to find the number of terms \( n \).

Again, using the nth term formula:
\[ a_n = -1 + (n-1) \cdot 5 = 34 \]

Let's solve these equations to find the values of \( d \) and \( n \).Here are the solutions for each problem:

**Problem 28:**
The common difference \( d \) is \( \frac{80}{9} \) or approximately \( 8.89 \).

**Problem 29:**
The common difference \( d \) is \( 2 \).

**Problem 30:**
The sequence has \( 8 \) terms.

Would you like to understand more about any of these problems or do you have any other questions?

Here are some follow-up topics you might consider:
1. How does changing the common difference \( d \) affect the sequence?
2. What would the sequence look like if you extended it to more terms?
3. How do you find the sum of the first \( n \) terms in an arithmetic sequence?
4. Can you have a negative common difference? What does it signify?
5. How does the formula for the nth term derive from the sequence definition?

**Tip:** Always verify your arithmetic sequence results by substituting back into the original formula to ensure consistency and correctness.

Solve for the indicated term based on the arithmetic sequences in the image provided.

Learn how to solve arithmetic sequence problems involving finding the common difference and term count. This problem covers key algebraic concepts and the arithmetic sequence formula. Suitable for Grades 9-11 students.

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