Let's solve the problem in the image.

Problem Breakdown:

We are tasked with finding a formula for the $n$-th term ($a_n$) of the given arithmetic sequence: $a_1 = 9, \, a_2 = 13, \, a_3 = 17, \, a_4 = 21, \, \dots$

Step 1: Identify the common difference ($d$)

The common difference $d$ in an arithmetic sequence is the difference between consecutive terms: $d = a_2 - a_1 = 13 - 9 = 4$

Step 2: Use the general formula for an arithmetic sequence

The general formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n - 1)d$

Step 3: Substitute $a_1 = 9$ and $d = 4$

$a_n = 9 + (n - 1)(4)$

Simplify: $a_n = 9 + 4n - 4$ $a_n = 4n + 5$

Final Formula:

$a_n = 4n + 5$

Answer:

The values to fill in the blanks are:

• $4$ (coefficient of $n$)
• $5$ (constant term)

Would you like a detailed explanation of any part of this process?

Related Questions:

1. What is the formula for the sum of the first $n$ terms in an arithmetic sequence?
2. How can we determine the term number $n$ when a specific value of $a_n$ is given?
3. What are the applications of arithmetic sequences in real-life scenarios?
4. How does the common difference affect the growth of the sequence?
5. What are the differences between arithmetic and geometric sequences?

Tip:

Always double-check your work by plugging in small values of $n$ to verify the derived formula against the sequence terms!