To find the explicit rule for the given geometric sequence, follow these steps:
1. Identify the first term [tex]\(a_1\)[/tex]:
- The first term is given as [tex]\(a_1 = 7\)[/tex].
2. Identify the common ratio [tex]\(r\)[/tex]:
- The recursive formula [tex]\(a_n = 13a_{n-1}\)[/tex] tells us that each term is obtained by multiplying the previous term by 13. Therefore, the common ratio [tex]\(r\)[/tex] is 13.
3. Recall the explicit formula for a geometric sequence:
- The general formula for the [tex]\(n\)[/tex]-th term of a geometric sequence with first term [tex]\(a_1\)[/tex] and common ratio [tex]\(r\)[/tex] is:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
4. Substitute the given values into the explicit formula:
- Here, [tex]\(a_1 = 7\)[/tex] and [tex]\(r = 13\)[/tex].
- Substitute these values into the geometric sequence formula:
[tex]\[
a_n = 7 \cdot 13^{n-1}
\][/tex]
5. Compare with the given options:
- Option A: [tex]\(a_n = 7 \cdot 13^{n-1}\)[/tex]
- Option B: [tex]\(a_n = 13 \cdot 7^{n-1}\)[/tex]
- Option C: [tex]\(a_n = 7 \cdot 13^{n+1}\)[/tex]
- Option D: [tex]\(a_n = 13 \cdot 7^{n+1}\)[/tex]
From the explicit formula derived, we see that it matches Option A.
Answer:
A. [tex]\(a_n = 7 \cdot 13^{n-1}\)[/tex]
