Let's analyze the given geometric sequence: [tex]\( 72, 36, 18, 9, \ldots \)[/tex].
A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio [tex]\( r \)[/tex].
First, we need to determine the common ratio [tex]\( r \)[/tex] for this sequence.
1. To find the common ratio [tex]\( r \)[/tex], we divide any term by the previous term:
[tex]\[
r = \frac{36}{72} = 0.5
\][/tex]
2. We can further confirm this by checking the ratio between other terms:
[tex]\[
r = \frac{18}{36} = 0.5
\][/tex]
[tex]\[
r = \frac{9}{18} = 0.5
\][/tex]
3. Therefore, our common ratio [tex]\( r \)[/tex] is [tex]\( 0.5 \)[/tex].
Next, we need to find the general formula for the [tex]\( n \)[/tex]-th term of a geometric sequence. The general formula for the [tex]\( n \)[/tex]-th term [tex]\( f(n) \)[/tex] of a geometric sequence with first term [tex]\( a \)[/tex] and common ratio [tex]\( r \)[/tex] is given by:
[tex]\[
f(n) = a \cdot r^{(n-1)}
\][/tex]
For our sequence:
- The first term [tex]\( a \)[/tex] is 72.
- The common ratio [tex]\( r \)[/tex] is [tex]\( 0.5 \)[/tex].
So, substituting these values into our formula, we get:
[tex]\[
f(n) = 72 \cdot (0.5)^{(n-1)}
\][/tex]
Therefore, the option that correctly represents the formula for the sequence is:
[tex]\[
f(n) = 72 \cdot (0.5)^{n-1}
\][/tex]
Thus, the correct answer is:
[tex]\[
f(n) = 72 \cdot (0.5)^{n-1}
\][/tex]
