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:
- \(a_n = 9n + 6\)