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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

Answer :

We are given the recursive relation

[tex]$$
f(n+1) = \frac{1}{3} f(n)
$$[/tex]

and we know that

[tex]$$
f(3) = 9.
$$[/tex]

Since the relation shows that each term is one third of the previous term, we can work backwards to find the earlier terms.

1. Start with [tex]$f(3)$[/tex]:

The relation for [tex]$n=2$[/tex] is

[tex]$$
f(3) = \frac{1}{3} f(2).
$$[/tex]

To solve for [tex]$f(2)$[/tex], multiply both sides by 3:

[tex]$$
f(2) = 3 \cdot f(3) = 3 \cdot 9 = 27.
$$[/tex]

2. Next, find [tex]$f(1)$[/tex]:

The relation for [tex]$n=1$[/tex] is

[tex]$$
f(2) = \frac{1}{3} f(1).
$$[/tex]

Again, multiply both sides by 3 to isolate [tex]$f(1)$[/tex]:

[tex]$$
f(1) = 3 \cdot f(2) = 3 \cdot 27 = 81.
$$[/tex]

Thus, the value of [tex]$f(1)$[/tex] is

[tex]$$
\boxed{81}.
$$[/tex]

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