Let's solve each part of the problem.

Part 4a

The given sequence is arithmetic, and we know the terms:

• $a_2 = 17$
• $a_3 = 29$
• $a_4 = 41$
• $a_7 = 77$

In an arithmetic sequence, each term is given by the formula: $a_n = a_1 + (n - 1)d$ where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Step 1: Find the common difference, $d$

Using $a_2 = 17$ and $a_3 = 29$: $d = a_3 - a_2 = 29 - 17 = 12$

Step 2: Determine $a_1$

Since $a_2 = a_1 + (2 - 1)d = a_1 + d$: $a_1 = a_2 - d = 17 - 12 = 5$

Step 3: Write the formula for $a_n$

Now we have $a_1 = 5$ and $d = 12$. The formula for the $n$-th term is: $a_n = 5 + (n - 1) \cdot 12$

So the answer for 4a is: $a_n = 5 + (n - 1) \cdot 12$

Part 4b

To find the 75th term ($a_{75}$): $a_{75} = 5 + (75 - 1) \cdot 12$ $a_{75} = 5 + 74 \cdot 12$ $a_{75} = 5 + 888$ $a_{75} = 893$

Therefore, the answer to 4b is:

• C. 893

Would you like further details or have any questions?

Here are five related questions that might help you expand on this concept:

1. How do you determine the common difference if more terms are given in an arithmetic sequence?
2. Can you derive the general term for any arithmetic sequence?
3. How do you find the sum of the first $n$ terms in an arithmetic sequence?
4. What changes in the formula if the sequence starts from $a_3$ instead of $a_1$?
5. How would you solve for $n$ if given $a_n$ and asked to find the position of a term?

Tip: Remember that in an arithmetic sequence, the difference between any two consecutive terms remains constant, which is key to solving these types of problems.