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Answer :
To solve this question, we need to use the recursive function provided to work backward from [tex]\( f(3) = 9 \)[/tex] to find [tex]\( f(1) \)[/tex].
The sequence is defined by the recursive relation [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], meaning each term is one-third of the previous term.
1. Start with [tex]\( f(3) \)[/tex]:
We know from the problem that [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can find [tex]\( f(2) \)[/tex] by reversing the operation:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Similarly, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], so to get [tex]\( f(1) \)[/tex], multiply [tex]\( f(2) \)[/tex] by 3:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
Thus, [tex]\( f(1) \)[/tex] is [tex]\( \boxed{81} \)[/tex].
The sequence is defined by the recursive relation [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], meaning each term is one-third of the previous term.
1. Start with [tex]\( f(3) \)[/tex]:
We know from the problem that [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can find [tex]\( f(2) \)[/tex] by reversing the operation:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Similarly, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], so to get [tex]\( f(1) \)[/tex], multiply [tex]\( f(2) \)[/tex] by 3:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
Thus, [tex]\( f(1) \)[/tex] is [tex]\( \boxed{81} \)[/tex].
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