To determine the explicit formula for the $n$-th term $a_n$ of this arithmetic sequence, let's analyze the given sequence:

The sequence is: $4, 9, 14, 19, \dots$

Step 1: Identify the first term and the common difference

1. First term ($a_1$): The first term is $4$.
2. Common difference ($d$): To find the common difference, subtract the first term from the second term: $d = 9 - 4 = 5$

Step 2: Write the formula for the $n$-th term of an arithmetic sequence

The general formula for the $n$-th term 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 position.

Step 3: Substitute the values

For this sequence:

• $a_1 = 4$
• $d = 5$

Thus, the formula becomes: $a_n = 4 + (n - 1) \cdot 5$

Step 4: Simplify the formula

Expanding this expression: $a_n = 4 + 5n - 5 = 5n - 1$

Final Answer

The explicit formula for the $n$-th term $a_n$ of the sequence is: $a_n = 5n - 1$

Would you like further details, or do you have any questions?

Here are five related questions that may help you explore arithmetic sequences further:

1. How do you find the sum of the first $n$ terms in an arithmetic sequence?
2. What are the properties that differentiate arithmetic sequences from geometric sequences?
3. How do you determine if a given sequence is arithmetic?
4. Can you derive the formula for the $n$-th term of an arithmetic sequence given any two terms?
5. How does the formula change if the common difference is negative?

Tip: For any arithmetic sequence, the common difference can be quickly found by subtracting any term from the following term.