To determine the explicit rule for the given geometric sequence, we need to follow these steps:
1. Identify the first term [tex]\(a\)[/tex]: The first term of the sequence [tex]\(f(n)\)[/tex] is given as [tex]\(3\)[/tex], so [tex]\(a = 3\)[/tex].
2. Determine the common ratio [tex]\(r\)[/tex]: The common ratio [tex]\(r\)[/tex] of a geometric sequence is found by dividing any term by the previous term. For this sequence, we can calculate the common ratio [tex]\(r\)[/tex] using the first two terms:
[tex]\[
r = \frac{f(2)}{f(1)} = \frac{15}{3} = 5
\][/tex]
3. Write the explicit rule: The general formula for the [tex]\(n\)[/tex]-th term of a geometric sequence is given by:
[tex]\[
f(n) = a \times r^{(n-1)}
\][/tex]
Using the values we have identified ([tex]\(a = 3\)[/tex] and [tex]\(r = 5\)[/tex]), the explicit rule for this sequence is:
[tex]\[
f(n) = 3 \times 5^{(n-1)}
\][/tex]
So, the explicit rule for the given geometric sequence is:
[tex]\[
f(n) = 3 \times 5^{(n-1)}
\][/tex]
