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]

Since the recursion relates a term to its previous term, we can rearrange the relation to express the previous term in terms of the next term:

[tex]$$
f(n) = 3 \cdot f(n+1).
$$[/tex]

To find [tex]$f(1)$[/tex], we first find [tex]$f(2)$[/tex] by setting [tex]$n = 2$[/tex]:

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

Next, we use [tex]$f(2)$[/tex] to calculate [tex]$f(1)$[/tex] by setting [tex]$n = 1$[/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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