Answer :
To find the explicit rule for the given geometric sequence, let's go through the process step-by-step:
The sequence given is:
60, 12, [tex]\(\frac{12}{5}\)[/tex], [tex]\(\frac{12}{25}\)[/tex], [tex]\(\frac{12}{125}\)[/tex], \ldots
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is 60.
2. Find the common ratio ([tex]\(r\)[/tex]):
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio ([tex]\(r\)[/tex]). To find [tex]\(r\)[/tex], divide the second term by the first term:
[tex]\[
r = \frac{12}{60} = 0.2
\][/tex]
3. Write the explicit formula:
The explicit formula for the [tex]\(n\)[/tex]th term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
Substituting the values of [tex]\(a_1\)[/tex] and [tex]\(r\)[/tex], we get:
[tex]\[
a_n = 60 \times (0.2)^{(n-1)}
\][/tex]
So, the explicit rule for the geometric sequence is:
[tex]\[
a_n = 60 \times (0.2)^{(n-1)}
\][/tex]
The sequence given is:
60, 12, [tex]\(\frac{12}{5}\)[/tex], [tex]\(\frac{12}{25}\)[/tex], [tex]\(\frac{12}{125}\)[/tex], \ldots
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is 60.
2. Find the common ratio ([tex]\(r\)[/tex]):
In a geometric sequence, each term is obtained by multiplying the previous term by a constant called the common ratio ([tex]\(r\)[/tex]). To find [tex]\(r\)[/tex], divide the second term by the first term:
[tex]\[
r = \frac{12}{60} = 0.2
\][/tex]
3. Write the explicit formula:
The explicit formula for the [tex]\(n\)[/tex]th term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
Substituting the values of [tex]\(a_1\)[/tex] and [tex]\(r\)[/tex], we get:
[tex]\[
a_n = 60 \times (0.2)^{(n-1)}
\][/tex]
So, the explicit rule for the geometric sequence is:
[tex]\[
a_n = 60 \times (0.2)^{(n-1)}
\][/tex]