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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][tex]$f(1)$[/tex][/tex]?

A. 1
B. 3
C. 27
D. 81

Answer :

To solve the problem, we are given a recursive function for a sequence: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. We also know that [tex]\( f(3) = 9 \)[/tex].

Our task is to find the value of [tex]\( f(1) \)[/tex].

1. Start from the known value:
- We have [tex]\( f(3) = 9 \)[/tex].

2. Use the recursive function to find [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can rearrange this to find [tex]\( f(n) = 3 \times f(n+1) \)[/tex].
- Therefore, [tex]\( f(2) = 3 \times f(3) \)[/tex].
- Substitute the known value: [tex]\( f(2) = 3 \times 9 = 27 \)[/tex].

3. Next, use the same formula to find [tex]\( f(1) \)[/tex]:
- Using the rearranged formula again, [tex]\( f(1) = 3 \times f(2) \)[/tex].
- Substitute the value we found for [tex]\( f(2) \)[/tex]: [tex]\( f(1) = 3 \times 27 = 81 \)[/tex].

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

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