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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 this problem, we need to find the value of [tex]\( f(1) \)[/tex] in the given sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], with the condition that [tex]\( f(3) = 9 \)[/tex].

We can determine [tex]\( f(1) \)[/tex] by working backward from the known value of [tex]\( f(3) \)[/tex].

1. Calculate [tex]\( f(2) \)[/tex]:
To find [tex]\( f(2) \)[/tex], we use the recursive definition. Given that [tex]\( f(3) = \frac{1}{3}f(2) \)[/tex], we can rearrange this to find [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 3 \times f(3)
\][/tex]
Substitute [tex]\( f(3) = 9 \)[/tex] into the equation:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]

2. Calculate [tex]\( f(1) \)[/tex]:
Now that we have [tex]\( f(2) \)[/tex], we can use it to find [tex]\( f(1) \)[/tex]. Again, using the recursive definition, [tex]\( f(2) = \frac{1}{3}f(1) \)[/tex], we rearrange to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]

Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].

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