Question

Write a recursive formula and an explicit formula for the following arithmetic sequence.
15​, 24​, 33​, 42​, 51​, ...
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Part 1
A recursive formula is a 1equals
  
15​, a Subscript nequals
  
a Subscript n minus 1 Baseline plus 9.
​(Simplify your​ answers.)
Part 2
An explicit formula is a Subscript nequals
  
enter your response here.

Answer 1

To analyze the arithmetic sequence \(15, 24, 33, 42, 51, \ldots\):

### Part 1: Recursive Formula

The first term \(a_1\) of the sequence is \(15\). The common difference \(d\) is calculated as follows:

\[
d = 24 - 15 = 9
\]

So, the recursive formula for the sequence is:

- \(a_1 = 15\)
- \(a_n = a_{n-1} + 9\) for \(n \geq 2\)

### Part 2: Explicit Formula

The explicit formula for an arithmetic sequence can be written as:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Substituting the known values (where \(a_1 = 15\) and \(d = 9\)):

\[
a_n = 15 + (n - 1) \cdot 9
\]

We can simplify this:

\[
a_n = 15 + 9n - 9
\]
\[
a_n = 9n + 6
\]

### Final Answers:

- **Recursive Formula**:
- \(a_1 = 15\)
- \(a_n = a_{n-1} + 9\) for \(n \geq 2\)

- **Explicit Formula**:
- \(a_n = 9n + 6\)