Sure! Let's determine the formula for the given geometric sequence step-by-step.
### Step 1: Identify the Common Ratio
The terms of the sequence are: [tex]\(72, 36, 18, 9, \ldots\)[/tex]
To find the common ratio [tex]\(r\)[/tex], we divide the second term by the first term:
[tex]\[ r = \frac{\text{second term}}{\text{first term}} = \frac{36}{72} = 0.5 \][/tex]
### Step 2: Write the General Formula for a Geometric Sequence
A geometric sequence with the first term [tex]\(a\)[/tex] and the common ratio [tex]\(r\)[/tex] can be written as:
[tex]\[ f(n) = a \cdot r^{n-1} \][/tex]
In this case, the first term [tex]\(a\)[/tex] is 72 and the common ratio [tex]\(r\)[/tex] is 0.5. Substituting these values into the general formula, we get:
[tex]\[ f(n) = 72 \cdot (0.5)^{n-1} \][/tex]
### Step 3: Match the Formula with the Given Options
We now compare our derived formula with the given options:
1. [tex]\( f(n) = 72(2)^{n-1} \)[/tex]
2. [tex]\( f(n) = 72(2)^{n+1} \)[/tex]
3. [tex]\( f(n) = 72(0.5)^{n-1} \)[/tex]
4. [tex]\( f(n) = 72(0.5)^{n+1} \)[/tex]
Our derived formula is:
[tex]\[ f(n) = 72 \cdot (0.5)^{n-1} \][/tex]
This matches the third option exactly:
[tex]\[ f(n) = 72(0.5)^{n-1} \][/tex]
### Conclusion
The correct formula for the given geometric sequence is:
[tex]\[ f(n) = 72(0.5)^{n-1} \][/tex]
