To determine the explicit function describing the given sequence [tex]\( 16, 24, 36, 54, \ldots \)[/tex], we need to identify the pattern of the sequence and derive a general formula that fits all the terms.
Step-by-Step Solution:
1. Examine the Sequence:
Let's list down the terms of the sequence:
- [tex]\( a_1 = 16 \)[/tex]
- [tex]\( a_2 = 24 \)[/tex]
- [tex]\( a_3 = 36 \)[/tex]
- [tex]\( a_4 = 54 \)[/tex]
2. Find the Ratios to Check for a Geometric Sequence:
Compute the ratio of successive terms to check if it forms a geometric sequence:
[tex]\[
\text{ratio}_1 = \frac{24}{16} = \frac{3}{2}
\][/tex]
[tex]\[
\text{ratio}_2 = \frac{36}{24} = \frac{3}{2}
\][/tex]
[tex]\[
\text{ratio}_3 = \frac{54}{36} = \frac{3}{2}
\][/tex]
Since the ratios are equal, the sequence is a geometric sequence.
3. Determine the Common Ratio and Initial Term:
From the ratios computed above, the common ratio [tex]\( r \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].
The initial term [tex]\( a \)[/tex] (when [tex]\( n = 1 \)[/tex]) is [tex]\( 16 \)[/tex].
4. Formulate the General Term for a Geometric Sequence:
The formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is given by:
[tex]\[
f(n) = a \cdot r^{(n-1)}
\][/tex]
Here, [tex]\( a = 16 \)[/tex] and [tex]\( r = \frac{3}{2} \)[/tex]. Plugging in these values, we get:
[tex]\[
f(n) = 16 \left(\frac{3}{2}\right)^{(n-1)}
\][/tex]
5. Match the Formula with Given Choices:
Among the given options, this matches with:
[tex]\[
B. \quad f(n) = 16 \left(\frac{3}{2}\right)^{(n-1)}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B}
\][/tex]
