Question

Write a recursive formula and an explicit formula for the following arithmetic sequence.
18​, 23​, 28​, 33​, 38​, ...
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Part 1
A recursive formula is a 1equals
  
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​(Simplify your​ answers.)

Answer 1

To determine a recursive formula and an explicit formula for the given arithmetic sequence \(18, 23, 28, 33, 38, \ldots\), we can analyze the sequence.

### Step 1: Identify the first term and common difference
- The first term \(a_1\) is \(18\).
- The common difference \(d\) is found by subtracting the first term from the second term:
\[
d = 23 - 18 = 5.
\]

### Step 2: Create the recursive formula
In a recursive formula, each term is defined in relation to the previous term. The general form of a recursive formula for an arithmetic sequence is:
\[
a_n = a_{n-1} + d
\]
where \(a_n\) is the \(n\)-th term, \(d\) is the common difference, and \(a_1\) is the first term.

Thus, the recursive formulas are:
- **First term:** \(a_1 = 18\)
- **Recursive relation:** \(a_n = a_{n-1} + 5\) for \(n \geq 2\)

### Step 3: Create the explicit formula
The explicit formula for an arithmetic sequence can be calculated using the formula:
\[
a_n = a_1 + (n-1) \cdot d
\]
Substituting the values of \(a_1\) and \(d\):
\[
a_n = 18 + (n-1) \cdot 5
\]
We can simplify this:
\[
a_n = 18 + 5n - 5 = 5n + 13.
\]

### Final Answers
Putting everything together:
- **Recursive Formula:**
\[
a_1 = 18, \quad a_n = a_{n-1} + 5 \text{ for } n \geq 2.
\]
- **Explicit Formula:**
\[
a_n = 5n + 13.
\]