To find both the recursive and explicit formulas for the given arithmetic sequence \( 15, 24, 33, 42, 51, \ldots \), we first identify the common difference in the sequence.
Common Difference
The difference between consecutive terms is:
- \( 24 - 15 = 9 \)
- \( 33 - 24 = 9 \)
- \( 42 - 33 = 9 \)
- \( 51 - 42 = 9 \)
The common difference \( d \) is \( 9 \).
Part 1: Recursive Formula
A recursive formula for an arithmetic sequence can be expressed as: \[ a_1 = 15 \quad \text{(the first term)} \] \[ a_n = a_{n-1} + 9 \quad \text{for } n \geq 2 \]
Recursive Formula
So the recursive formula is: \[ a_1 = 15, \quad a_n = a_{n-1} + 9 \text{ for } n \geq 2 \]
Part 2: Explicit Formula
The explicit formula for an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1)d \] Where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.
Substituting the values we have:
- \( a_1 = 15 \)
- \( d = 9 \)
Thus, the explicit formula is: \[ a_n = 15 + (n - 1) \cdot 9 \] Simplifying this: \[ a_n = 15 + 9n - 9 \] \[ a_n = 9n + 6 \]
Explicit Formula
So, the explicit formula is: \[ a_n = 9n + 6 \]
Final Answers
- Recursive formula: \( a_1 = 15, \quad a_n = a_{n-1} + 9 \text{ for } n \geq 2 \)
- Explicit formula: \( a_n = 9n + 6 \)