To determine the explicit rule for a geometric sequence, we need to use the general form of the nth term of a geometric sequence, which is given by [tex]\( a_n = a \cdot r^{n-1} \)[/tex], where:
- [tex]\( a \)[/tex] is the first term of the sequence
- [tex]\( r \)[/tex] is the common ratio
- [tex]\( n \)[/tex] is the term number
Given the first few terms of the sequence are [tex]\( 4.05, 1.35, 0.45, 0.15, \ldots \)[/tex]:
1. Identifying the First Term:
The first term [tex]\( a \)[/tex] is clearly [tex]\( 4.05 \)[/tex].
2. Determining the Common Ratio:
To find the common ratio [tex]\( r \)[/tex], divide the second term by the first term:
[tex]\[
r = \frac{1.35}{4.05}
\][/tex]
Simplifying this:
[tex]\[
r = \frac{1}{3} \approx 0.3333
\][/tex]
3. Finding the Explicit Rule:
Now, substitute [tex]\( a = 4.05 \)[/tex] and [tex]\( r = \frac{1}{3} \)[/tex] into the geometric sequence formula [tex]\( a_n = a \cdot r^{n-1} \)[/tex]:
[tex]\[
a_n = 4.05 \cdot \left(\frac{1}{3}\right)^{n-1}
\][/tex]
Therefore, the explicit rule for the geometric sequence is:
[tex]\[
a_n = 4.05 \left( \frac{1}{3} \right)^{n-1}
\][/tex]
Thus, the correct option is:
[tex]\[
a_n = 4.05 \left( \frac{1}{3} \right)^{n-1}
\][/tex]
