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 relation
[tex]$$
f(n+1)=\frac{1}{3}f(n)
$$[/tex]
and the value
[tex]$$
f(3)=9.
$$[/tex]

To find [tex]$f(1)$[/tex], we can work backwards from [tex]$f(3)$[/tex].

1. First, express [tex]$f(3)$[/tex] in terms of [tex]$f(2)$[/tex]:
[tex]$$
f(3)=\frac{1}{3}f(2).
$$[/tex]
Solving for [tex]$f(2)$[/tex] gives:
[tex]$$
f(2)=3f(3)=3\cdot 9=27.
$$[/tex]

2. Next, express [tex]$f(2)$[/tex] in terms of [tex]$f(1)$[/tex]:
[tex]$$
f(2)=\frac{1}{3}f(1).
$$[/tex]
Solving for [tex]$f(1)$[/tex] gives:
[tex]$$
f(1)=3f(2)=3\cdot 27=81.
$$[/tex]

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

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

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