Answer :
Sure, let's solve this step by step.
We are given the recursive function:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we know a specific value:
[tex]\[ f(3) = 9 \][/tex]
We need to determine the value of [tex]\( f(1) \)[/tex].
First, let's find [tex]\( f(2) \)[/tex] using the recursive function. We know:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Rearranging this to solve for [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 3 \times f(3) \][/tex]
Since [tex]\( f(3) = 9 \)[/tex]:
[tex]\[ f(2) = 3 \times 9 = 27 \][/tex]
Next, let's find [tex]\( f(1) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Rearranging this to solve for [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 3 \times f(2) \][/tex]
Since [tex]\( f(2) = 27 \)[/tex]:
[tex]\[ f(1) = 3 \times 27 = 81 \][/tex]
So, the value of [tex]\( f(1) \)[/tex] is:
[tex]\[ \boxed{81} \][/tex]
We are given the recursive function:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we know a specific value:
[tex]\[ f(3) = 9 \][/tex]
We need to determine the value of [tex]\( f(1) \)[/tex].
First, let's find [tex]\( f(2) \)[/tex] using the recursive function. We know:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Rearranging this to solve for [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 3 \times f(3) \][/tex]
Since [tex]\( f(3) = 9 \)[/tex]:
[tex]\[ f(2) = 3 \times 9 = 27 \][/tex]
Next, let's find [tex]\( f(1) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Rearranging this to solve for [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 3 \times f(2) \][/tex]
Since [tex]\( f(2) = 27 \)[/tex]:
[tex]\[ f(1) = 3 \times 27 = 81 \][/tex]
So, the value of [tex]\( f(1) \)[/tex] is:
[tex]\[ \boxed{81} \][/tex]
