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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 :

To solve the problem, let's work through the sequence using the recursive function provided:

We are given that the sequence is defined by the function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that each term of the sequence is one-third of the previous term.

We also know that [tex]\( f(3) = 9 \)[/tex] and we need to find the initial term of the sequence, [tex]\( f(0) \)[/tex].

Let's work backwards from the known term:

1. Since [tex]\( f(3) = 9 \)[/tex], we can find [tex]\( f(2) \)[/tex] by recognizing that:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex] gives:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

2. Now, using [tex]\( f(2) = 27 \)[/tex], we find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex] gives:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

3. Finally, using [tex]\( f(1) = 81 \)[/tex], we find [tex]\( f(0) \)[/tex]:
[tex]\[
f(1) = \frac{1}{3} f(0)
\][/tex]
Solving for [tex]\( f(0) \)[/tex] gives:
[tex]\[
f(0) = 81 \times 3 = 243
\][/tex]

Therefore, the initial term [tex]\( f(0) \)[/tex] of the sequence is [tex]\(\boxed{243}\)[/tex].

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