The problem involves examining four different sequences, each of which has its own unique recursive equation and initial term. Let’s break down each one step-by-step to identify the detailed recursive format specified.
## Sequence Analysis:
1. First Sequence:
- Recursive equation: [tex]\( a_n = 50 \cdot a_{n-1} \)[/tex]
- Initial term: [tex]\( a_1 = 3 \)[/tex]
This describes a geometric sequence where each term is obtained by multiplying the previous term by 50.
Sequence format: [tex]\( a_n = 50 \cdot a_{n-1} \)[/tex], with [tex]\( a_1 = 3 \)[/tex]
2. Second Sequence:
- Recursive equation: [tex]\( a_n = a_{n-1} + 3 \)[/tex]
- Initial term: [tex]\( a_1 = 50 \)[/tex]
This describes an arithmetic sequence where each term is obtained by adding 3 to the previous term.
Sequence format: [tex]\( a_n = a_{n-1} + 3 \)[/tex], with [tex]\( a_1 = 50 \)[/tex]
3. Third Sequence:
- Recursive equation: [tex]\( a_n = a_{n-1} + 50 \)[/tex]
- Initial term: [tex]\( a_1 = 3 \)[/tex]
This describes an arithmetic sequence where each term is obtained by adding 50 to the previous term.
Sequence format: [tex]\( a_n = a_{n-1} + 50 \)[/tex], with [tex]\( a_1 = 3 \)[/tex]
4. Fourth Sequence:
- Recursive equation: [tex]\( a_n = 3 \cdot a_{n-1} \)[/tex]
- Initial term: [tex]\( a_1 = 50 \)[/tex]
This describes a geometric sequence where each term is obtained by multiplying the previous term by 3.
Sequence format: [tex]\( a_n = 3 \cdot a_{n-1} \)[/tex], with [tex]\( a_1 = 50 \)[/tex]
## Final Summary:
Given the sequences, the respective recursive equations and initial terms fit the following patterns:
1. [tex]\( a_n = 50 \cdot a_{n-1} \)[/tex], with [tex]\( a_1 = 3 \)[/tex]
2. [tex]\( a_n = a_{n-1} + 3 \)[/tex], with [tex]\( a_1 = 50 \)[/tex]
3. [tex]\( a_n = a_{n-1} + 50 \)[/tex], with [tex]\( a_1 = 3 \)[/tex]
4. [tex]\( a_n = 3 \cdot a_{n-1} \)[/tex], with [tex]\( a_1 = 50 \)[/tex]
Thus, the recursive equations accurately reflect the format provided by the problems.
