I see you've uploaded a math problem that involves arithmetic sequences and series. Let's go through the questions one by one:

Problem 5:

An arithmetic sequence has a 10th term of 17 and a 14th term of 30. Find the common difference.

For an arithmetic sequence, we can use the formula for the $n$-th 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,
• $n$ is the term number.

You know:

• $a_{10} = 17$,
• $a_{14} = 30$.

We can use the formula for these two terms to create two equations, then solve for the common difference $d$.

Problem 6:

An arithmetic sequence has a 7th term of 54 and a 13th term of 94. Find the common difference.

We can follow the same process as Problem 5:

• $a_7 = 54$,
• $a_{13} = 94$.

Set up the equations based on the general formula for the $n$-th term of an arithmetic sequence and solve for $d$.

Problem 7:

Find the sum of the positive terms of the arithmetic sequence $85, 78, 71, \ldots$

Here, you need to:

1. Identify the common difference $d = -7$.
2. Use the formula for the sum of the first $n$ terms of an arithmetic sequence:

$S_n = \frac{n}{2} \left(a_1 + a_n \right)$

You have $a_1 = 85$ and the sequence decreases. Find how many terms are positive by finding the last positive term.

Problem 8:

A theater has 32 rows of seats. If there are 26 seats in the 1st row, 30 in the 2nd, 34 in the 3rd, and so on, how many seats are there in total?

This is also an arithmetic sequence with:

• $a_1 = 26$,
• Common difference $d = 4$,
• The total number of rows $n = 32$.

Use the sum formula for the arithmetic sequence to calculate the total number of seats.

Let me know if you'd like detailed solutions for each of these problems, or if you have any other questions!