To determine the explicit rule for the given geometric sequence:
[tex]\[ 60, 12, \frac{12}{5}, \frac{12}{25}, \frac{12}{125}, \ldots \][/tex]
let's follow these steps:
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is [tex]\(a_1 = 60\)[/tex].
2. Determine the common ratio ([tex]\(r\)[/tex]):
The common ratio [tex]\(r\)[/tex] can be found by dividing any term by its preceding term.
[tex]\[
r = \frac{12}{60} = 0.2
\][/tex]
3. Use the explicit formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
The formula for the [tex]\(n\)[/tex]-th term is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where:
- [tex]\(a_1\)[/tex] is the first term.
- [tex]\(r\)[/tex] is the common ratio.
- [tex]\(n\)[/tex] is the term number.
Plugging in the values we have:
- [tex]\(a_1 = 60\)[/tex]
- [tex]\(r = 0.2\)[/tex]
The explicit rule for the [tex]\(n\)[/tex]-th term of this geometric sequence is:
[tex]\[
a_n = 60 \cdot (0.2)^{(n-1)}
\][/tex]
