To find the explicit rule of a geometric sequence given a recursive formula and the first term, we can follow these steps:
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common ratio ([tex]\(r\)[/tex]):
- The first term [tex]\(a_1\)[/tex] is given as 7.
- The recursive formula [tex]\(a_n = 13 \cdot a_{n-1}\)[/tex] indicates that the common ratio [tex]\(r\)[/tex] is 13.
2. Recall the general form for the explicit rule of a geometric sequence:
The n-th term of a geometric sequence can be expressed 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 given first term and common ratio into the general form:
- First term [tex]\(a_1 = 7\)[/tex]
- Common ratio [tex]\(r = 13\)[/tex]
Therefore, the explicit formula becomes:
[tex]\[
a_n = 7 \cdot 13^{n-1}
\][/tex]
4. Match this explicit formula to the provided options:
Let's compare the derived formula [tex]\(a_n = 7 \cdot 13^{n-1}\)[/tex] with the given options:
- Option A: [tex]\(a_n = 13 \cdot 7^{n-1}\)[/tex]
- Option B: [tex]\(a_n = 7 \cdot 13^{n+1}\)[/tex]
- Option C: [tex]\(a_n = 7 \cdot 13^{n-1}\)[/tex]
- Option D: [tex]\(a_n = 13 \cdot 7^{n+1}\)[/tex]
We see that Option C matches exactly with our derived explicit rule.
Therefore, the correct answer is:
C. [tex]\(a_n = 7 \cdot 13^{n-1}\)[/tex]
