To determine the formula that represents the given geometric sequence [tex]\(72, 36, 18, 9, \ldots\)[/tex], let's analyze it step-by-step.
### Identify the First Term and Common Ratio
1. First Term ([tex]\(a\)[/tex]):
The first term of the sequence is clearly given as [tex]\(a = 72\)[/tex].
2. Common Ratio ([tex]\(r\)[/tex]):
To find the common ratio, we observe the relationship between consecutive terms.
[tex]\[
r = \frac{36}{72} = 0.5
\][/tex]
This means each term is obtained by multiplying the previous term by 0.5.
### General Formula for Geometric Sequence
The general formula for a geometric sequence is given by:
[tex]\[
f(n) = a \cdot r^{(n-1)}
\][/tex]
Substituting the identified values:
- [tex]\(a = 72\)[/tex]
- [tex]\(r = 0.5\)[/tex]
We have:
[tex]\[
f(n) = 72 \cdot (0.5)^{(n-1)}
\][/tex]
### Conclusion
Thus, the formula that best represents the given geometric sequence is:
[tex]\[
f(n) = 72 \cdot (0.5)^{(n-1)}
\][/tex]
So, the correct option is:
\[
f(n) = 72 \cdot (0.5)^{n-1}
\
