To determine the correct formula for the geometric sequence given with the terms [tex]\(72, 36, 18, 9, \ldots\)[/tex], let's go through a detailed solution:
1. Identify the first term and common ratio:
- The first term [tex]\( a_1 \)[/tex] is [tex]\( 72 \)[/tex].
- The second term [tex]\( a_2 \)[/tex] is [tex]\( 36 \)[/tex].
- To find the common ratio [tex]\( r \)[/tex], we divide the second term by the first term:
[tex]\[
r = \frac{a_2}{a_1} = \frac{36}{72} = 0.5
\][/tex]
2. General Formula for a Geometric Sequence:
- The nth term [tex]\( f(n) \)[/tex] of a geometric sequence can be written as:
[tex]\[
f(n) = a \cdot r^{(n-1)}
\][/tex]
where [tex]\( a \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
3. Substitute the values for this sequence:
- Here, [tex]\( a = 72 \)[/tex] and [tex]\( r = 0.5 \)[/tex]. Plugging these values into the formula:
[tex]\[
f(n) = 72 \cdot (0.5)^{(n-1)}
\][/tex]
After deriving the formula, we compare it with the provided options:
- [tex]\( f(n) = 72 (2)^{n-1} \)[/tex]
- [tex]\( f(n) = 72 (2)^{n+1} \)[/tex]
- [tex]\( f(n) = 72 (0.5)^{2-1} \)[/tex]
- [tex]\( f(n) = 72 (0.5)^{n+1} \)[/tex]
Clearly, the correct option that matches our derived formula [tex]\( f(n) = \ 72 \cdot (0.5)^{(n-1)} \)[/tex] is not directly listed among the provided options.
However, to match the correct format:
- Correct formula interpretation: It's evident that our derived formula is [tex]\( f(n) = 72 \cdot (0.5)^{(n-1)} \)[/tex].
Matching closely with our derived result, the correct answer derived from our formula is:
[tex]\[
f(n) = 72 \cdot (0.5)^{n-1}
\][/tex]
Therefore, translating it into given options, the correct response is:
The provided answer choice most closely aligns with our direct interpretation. Thus, the best correct formula is:
[tex]\[
f(n) = 72 \cdot (0.5)^{n-1}
\][/tex]
