Answer :
To find the explicit rule for the given geometric sequence, we need to identify the first term and the common ratio of the sequence.
1. Identify the First Term:
The first term of the sequence is 60.
2. Determine the Common Ratio:
The common ratio ([tex]\( r \)[/tex]) is found by dividing any term by the previous term. We can choose the second term and divide it by the first term:
[tex]\[
r = \frac{12}{60} = \frac{1}{5}
\][/tex]
3. Write the Explicit Formula:
The explicit formula for the [tex]\( n \)[/tex]-th term of a geometric sequence with the first term [tex]\( a_1 \)[/tex] and common ratio [tex]\( r \)[/tex] is given by:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
For our sequence, this becomes:
[tex]\[
a_n = 60 \times \left(\frac{1}{5}\right)^{n-1}
\][/tex]
Therefore, the explicit rule for the geometric sequence is:
[tex]\[
a_n = 60 \times \left(\frac{1}{5}\right)^{n-1}
\][/tex]
1. Identify the First Term:
The first term of the sequence is 60.
2. Determine the Common Ratio:
The common ratio ([tex]\( r \)[/tex]) is found by dividing any term by the previous term. We can choose the second term and divide it by the first term:
[tex]\[
r = \frac{12}{60} = \frac{1}{5}
\][/tex]
3. Write the Explicit Formula:
The explicit formula for the [tex]\( n \)[/tex]-th term of a geometric sequence with the first term [tex]\( a_1 \)[/tex] and common ratio [tex]\( r \)[/tex] is given by:
[tex]\[
a_n = a_1 \times r^{(n-1)}
\][/tex]
For our sequence, this becomes:
[tex]\[
a_n = 60 \times \left(\frac{1}{5}\right)^{n-1}
\][/tex]
Therefore, the explicit rule for the geometric sequence is:
[tex]\[
a_n = 60 \times \left(\frac{1}{5}\right)^{n-1}
\][/tex]