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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. 27
B. 1
C. 3
D. 81

Answer :

We are given the recursive sequence defined by

$$
f(n+1) = \frac{1}{3} f(n).
$$

This can be rewritten as

$$
f(n) = 3 f(n+1).
$$

We know that

$$
f(3) = 9.
$$

Step 1: Compute $f(2)$
Since

$$
f(3) = \frac{1}{3} f(2),
$$

we can solve for $f(2)$:

$$
f(2) = 3 f(3) = 3 \times 9 = 27.
$$

Step 2: Compute $f(1)$
Similarly, using the relation

$$
f(2) = \frac{1}{3} f(1),
$$

we solve for $f(1)$:

$$
f(1) = 3 f(2) = 3 \times 27 = 81.
$$

Thus, the value of $f(1)$ is

$$
\boxed{81}.
$$

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