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------------------------------------------------ A sequence is defined by the recursive function [tex]f(n+1)=\frac{1}{3} f(n)[/tex]. If [tex]f(3)=9[/tex], what is [tex]f(1)[/tex]?

A. 1
B. 3
C. 27
D. 81

To solve the problem, we need to find the value of [tex]$ f(1) $[/tex] for the given recursive sequence. The sequence is defined as [tex]$ f(n+1) = \frac{1}{3} f(n) $[/tex]. We also know that [tex]$ f(3) = 9 $[/tex].

Let's work backwards from the information provided:

1. Find [tex]$ f(2) $[/tex]:

We have the equation:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute [tex]$ f(3) = 9 $[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]$ f(2) $[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

2. Find [tex]$ f(1) $[/tex]:

Next, use the equation:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute [tex]$ f(2) = 27 $[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To solve for [tex]$ f(1) $[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

Therefore, the value of [tex]$ f(1) $[/tex] is 81.