Answer :
To solve the problem, we need to determine the value of [tex]\( f(1) \)[/tex] using the recursive function defined as [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and the information that [tex]\( f(3) = 9 \)[/tex].
Since we are given [tex]\( f(3) = 9 \)[/tex] and need to find [tex]\( f(1) \)[/tex], we will work backwards using the recursive relationship. The recursive formula tells us how to go from [tex]\( f(n) \)[/tex] to [tex]\( f(n+1) \)[/tex]. To find the previous terms, we need to reverse this relation.
The recursive formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] can be rearranged to find [tex]\( f(n) \)[/tex] when [tex]\( f(n+1) \)[/tex] is known:
[tex]\[ f(n) = 3 \times f(n+1) \][/tex]
Let's use this to calculate the values step by step:
1. Start with the known value: [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex] using [tex]\( f(n) = 3 \times f(n+1) \)[/tex]:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Next, find [tex]\( f(1) \)[/tex]:
[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].
Since we are given [tex]\( f(3) = 9 \)[/tex] and need to find [tex]\( f(1) \)[/tex], we will work backwards using the recursive relationship. The recursive formula tells us how to go from [tex]\( f(n) \)[/tex] to [tex]\( f(n+1) \)[/tex]. To find the previous terms, we need to reverse this relation.
The recursive formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] can be rearranged to find [tex]\( f(n) \)[/tex] when [tex]\( f(n+1) \)[/tex] is known:
[tex]\[ f(n) = 3 \times f(n+1) \][/tex]
Let's use this to calculate the values step by step:
1. Start with the known value: [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex] using [tex]\( f(n) = 3 \times f(n+1) \)[/tex]:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Next, find [tex]\( f(1) \)[/tex]:
[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].
