To solve this, let's break down the arithmetic sequence step by step:

Step 1: Determine the common difference

The sequence provided is: $33, 39, 45, \ldots$ We can find the common difference by subtracting the first term from the second: $39 - 33 = 6$ Similarly: $45 - 39 = 6$ Thus, the common difference $d = 6$.

Step 2: General form of an arithmetic sequence

The general form of an arithmetic sequence is: $a(n) = a_1 + (n - 1)d$ where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Here, $a_1 = 33$ and $d = 6$. So, the function becomes: $a(n) = 33 + (n - 1) \cdot 6$ Simplifying this: $a(n) = 33 + 6n - 6$ $a(n) = 6n + 27$

Step 3: Select the correct function notation

Thus, the correct function notation is: $\boxed{b) \ a(n) = 6n + 27}$

Step 4: Find the 10th term

To find the 10th term, substitute $n = 10$ into the function: $a(10) = 6(10) + 27 = 60 + 27 = 87$

So, the 10th term of the sequence is: $\boxed{87}$

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Would you like more details on any of these steps?

Here are 5 related questions:

1. How do you find the nth term of a general arithmetic sequence?
2. What is the sum of the first 10 terms of this sequence?
3. How can you determine the number of terms in an arithmetic sequence if the last term is known?
4. How does the common difference affect the growth of an arithmetic sequence?
5. What is the relationship between arithmetic sequences and linear functions?

Tip: Always check for the common difference in an arithmetic sequence before proceeding with the formula.