To determine the explicit rule for the geometric sequence, let's break down the information given and follow the logical steps to find the correct expression.
### Given Information:
1. Recursive Formula: [tex]\( a_n = 5 \cdot a_{n-1} \)[/tex]
2. First Term: [tex]\( a_1 = 23 \)[/tex]
### Step-by-Step Solution:
1. Identify the General Form of a Geometric Sequence:
A geometric sequence can be defined recursively by:
[tex]\[
a_n = r \cdot a_{n-1}
\][/tex]
Here, [tex]\( r \)[/tex] is the common ratio. In our problem, [tex]\( r = 5 \)[/tex].
2. Determine How to Convert the Recursive Formula to an Explicit Formula:
For a geometric sequence, the explicit formula is generally written as:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
where [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
3. Substitute the Known Values:
- [tex]\( a_1 = 23 \)[/tex]
- [tex]\( r = 5 \)[/tex]
Therefore, the explicit formula becomes:
[tex]\[
a_n = 23 \cdot 5^{n-1}
\][/tex]
4. Compare with the Given Options:
- Option A: [tex]\( 23 \cdot 5^{n-1} \)[/tex]
- Option B: [tex]\( 5 \cdot 23^{n+1} \)[/tex]
- Option C: [tex]\( 23 \cdot 5^{n+1} \)[/tex]
- Option D: [tex]\( 5 \cdot 23^{n-1} \)[/tex]
The option that matches our explicit formula is:
- Option A: [tex]\( 23 \cdot 5^{n-1} \)[/tex]
### Final Answer:
The explicit rule for the given geometric sequence is Option A. Hence, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
