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Which of the following is the explicit rule for a geometric sequence defined by a recursive formula of [tex]$a_n = 13 a_{n-1}$[/tex] for which the first term is 7?

A. [tex]$a_n = 7 \cdot 13^{n-1}$[/tex]
B. [tex][tex]$a_n = 13 \cdot 7^{n-1}$[/tex][/tex]
C. [tex]$a_n = 7 \cdot 13^{n+1}$[/tex]
D. [tex]$a_n = 13 \cdot 7^{n+1}$[/tex]


Sagot :

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]
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