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Answer :
Sure! Let's solve the problem step-by-step, using the recursive sequence provided:
We have the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This tells us how each term in the sequence is related to the previous one. If we know one term, we can find others by either moving forward or backward using this relationship.
Given:
[tex]\[ f(3) = 9 \][/tex]
We need to find [tex]\( f(1) \)[/tex]. To do this, we'll work backwards from [tex]\( f(3) \)[/tex] to [tex]\( f(1) \)[/tex].
### Step 1: Determine [tex]\( f(2) \)[/tex]
The relationship [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] can be rearranged to find a previous term:
[tex]\[ f(n) = 3 \times f(n+1) \][/tex]
To find [tex]\( f(2) \)[/tex] from [tex]\( f(3) \)[/tex]:
[tex]\[ f(2) = 3 \times f(3) = 3 \times 9 = 27 \][/tex]
### Step 2: Determine [tex]\( f(1) \)[/tex]
Using the same relationship, we find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[ f(1) = 3 \times f(2) = 3 \times 27 = 81 \][/tex]
Therefore, [tex]\( f(1) = 81 \)[/tex].
The answer to the question is 81.
We have the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This tells us how each term in the sequence is related to the previous one. If we know one term, we can find others by either moving forward or backward using this relationship.
Given:
[tex]\[ f(3) = 9 \][/tex]
We need to find [tex]\( f(1) \)[/tex]. To do this, we'll work backwards from [tex]\( f(3) \)[/tex] to [tex]\( f(1) \)[/tex].
### Step 1: Determine [tex]\( f(2) \)[/tex]
The relationship [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] can be rearranged to find a previous term:
[tex]\[ f(n) = 3 \times f(n+1) \][/tex]
To find [tex]\( f(2) \)[/tex] from [tex]\( f(3) \)[/tex]:
[tex]\[ f(2) = 3 \times f(3) = 3 \times 9 = 27 \][/tex]
### Step 2: Determine [tex]\( f(1) \)[/tex]
Using the same relationship, we find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[ f(1) = 3 \times f(2) = 3 \times 27 = 81 \][/tex]
Therefore, [tex]\( f(1) = 81 \)[/tex].
The answer to the question is 81.
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