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Answer :
To solve this problem, we will use the given recursive relationship to work backwards starting from [tex]\( f(3) = 9 \)[/tex] to find [tex]\( f(1) \)[/tex].
The sequence is defined by the formula:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
1. Start with [tex]\( f(3) \)[/tex]:
We know [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
Using the recursive relationship backwards, we need to find [tex]\( f(2) \)[/tex] such that:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Similarly, we find [tex]\( f(1) \)[/tex] using:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex], we multiply both sides by 3:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
The sequence is defined by the formula:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
1. Start with [tex]\( f(3) \)[/tex]:
We know [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
Using the recursive relationship backwards, we need to find [tex]\( f(2) \)[/tex] such that:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Similarly, we find [tex]\( f(1) \)[/tex] using:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex], we multiply both sides by 3:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
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