To determine the explicit formula for the given sequence with a common ratio of [tex]\(\frac{3}{2}\)[/tex] and [tex]\(f(5) = 81\)[/tex], follow these steps:
1. Identify the General Formula of a Geometric Sequence:
The general formula for the [tex]\(n\)[/tex]-th term of a geometric sequence is:
[tex]\[
f(n) = a \cdot r^{n-1}
\][/tex]
where [tex]\(a\)[/tex] is the first term (also called the initial term) and [tex]\(r\)[/tex] is the common ratio.
2. Substitute the Given Information into the Formula:
We are given:
- The common ratio [tex]\(r = \frac{3}{2}\)[/tex]
- [tex]\(f(5) = 81\)[/tex]
Let's substitute these into the geometric sequence formula:
[tex]\[
f(5) = a \cdot \left(\frac{3}{2}\right)^{5-1} = a \cdot \left(\frac{3}{2}\right)^4
\][/tex]
3. Solve for the First Term [tex]\(a\)[/tex]:
Given that [tex]\(f(5) = 81\)[/tex], we can write the equation as:
[tex]\[
81 = a \cdot \left(\frac{3}{2}\right)^4
\][/tex]
We need to isolate [tex]\(a\)[/tex]. First, calculate [tex]\(\left(\frac{3}{2}\right)^4\)[/tex]:
[tex]\[
\left(\frac{3}{2}\right)^4 = \frac{3^4}{2^4} = \frac{81}{16}
\][/tex]
Substitute back into the equation:
[tex]\[
81 = a \cdot \frac{81}{16}
\][/tex]
Now, solve for [tex]\(a\)[/tex] by multiplying both sides by [tex]\(\frac{16}{81}\)[/tex]:
[tex]\[
a = 81 \cdot \frac{16}{81} = 16
\][/tex]
4. Write the Explicit Formula:
Now that we have determined [tex]\(a = 16\)[/tex] and [tex]\(r = \frac{3}{2}\)[/tex], the explicit formula for the sequence is:
[tex]\[
f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}
\][/tex]
Thus, the correct formula from the given options is:
[tex]\[
\boxed{f(x) = 16 \left(\frac{3}{2}\right)^{x-1}}
\][/tex]
