To determine the explicit rule for the given geometric sequence [tex]\(120, 40, \frac{40}{3}, \frac{40}{9}, \frac{40}{27}, \ldots\)[/tex]:
1. Identify the First Term ([tex]\(a_1\)[/tex]):
The first term of the sequence is [tex]\(a_1 = 120\)[/tex].
2. Determine the Common Ratio ([tex]\(r\)[/tex]):
The common ratio of a geometric sequence can be found by dividing any term by the previous term.
[tex]\[
r = \frac{\text{second term}}{\text{first term}} = \frac{40}{120} = \frac{1}{3}
\][/tex]
3. Formulate the General Rule for the [tex]\(n\)[/tex]th Term ([tex]\(a_n\)[/tex]):
The general formula for the [tex]\(n\)[/tex]th term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Substituting the values we have found, the first term [tex]\(a_1 = 120\)[/tex] and the common ratio [tex]\(r = \frac{1}{3}\)[/tex]:
[tex]\[
a_n = 120 \cdot \left(\frac{1}{3}\right)^{(n-1)}
\][/tex]
So, the explicit rule for the given geometric sequence is:
[tex]\[
a_n = 120 \cdot \left(\frac{1}{3}\right)^{(n-1)}
\][/tex]
