College

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

Answer :

We are given the recursive relation

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

and the value

$$
f(3) = 9.
$$

To find $f(1)$, we work backwards:

1. First, express $f(3)$ in terms of $f(2)$:

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

Solving for $f(2)$, multiply both sides by 3:

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

2. Next, express $f(2)$ in terms of $f(1)$:

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

Again, solving for $f(1)$, multiply both sides by 3:

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

Thus, the value of $f(1)$ is $\boxed{81}$.