A geometric sequence begins with [tex]\(72, 36, 18, 9, \ldots\)[/tex]

Which option below represents the formula for the sequence?

A. [tex]\(f(n) = 72(2)^{n-1}\)[/tex]
B. [tex]\(f(n) = 72(2)^{n+1}\)[/tex]
C. [tex]\(f(n) = 72(0.5)^{n-1}\)[/tex]
D. [tex]\(f(n) = 72(0.5)^{n+1}\)[/tex]



Answer :

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}
\