High School

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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 function

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

and the value

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

Since the function decreases by a factor of [tex]$\frac{1}{3}$[/tex] as [tex]$n$[/tex] increases by 1, we can reverse the process. To find [tex]$f(1)$[/tex], we first find [tex]$f(2)$[/tex] using the reverse recurrence relation:

[tex]$$
f(3) = \frac{1}{3} f(2) \quad \Longrightarrow \quad f(2) = 3 \cdot f(3) = 3 \cdot 9 = 27.
$$[/tex]

Next, we find [tex]$f(1)$[/tex] similarly:

[tex]$$
f(2) = \frac{1}{3} f(1) \quad \Longrightarrow \quad f(1) = 3 \cdot f(2) = 3 \cdot 27 = 81.
$$[/tex]

Thus, the answer is

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

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