Answer :
We are given the recursive function
[tex]$$
f(n+1) = \frac{1}{3} f(n)
$$[/tex]
and the value
[tex]$$
f(3) = 9.
$$[/tex]
Since the function decreases by a factor of [tex]$\frac{1}{3}$[/tex] as [tex]$n$[/tex] increases by 1, we can reverse the process. To find [tex]$f(1)$[/tex], we first find [tex]$f(2)$[/tex] using the reverse recurrence relation:
[tex]$$
f(3) = \frac{1}{3} f(2) \quad \Longrightarrow \quad f(2) = 3 \cdot f(3) = 3 \cdot 9 = 27.
$$[/tex]
Next, we find [tex]$f(1)$[/tex] similarly:
[tex]$$
f(2) = \frac{1}{3} f(1) \quad \Longrightarrow \quad f(1) = 3 \cdot f(2) = 3 \cdot 27 = 81.
$$[/tex]
Thus, the answer is
[tex]$$
\boxed{81}.
$$[/tex]
[tex]$$
f(n+1) = \frac{1}{3} f(n)
$$[/tex]
and the value
[tex]$$
f(3) = 9.
$$[/tex]
Since the function decreases by a factor of [tex]$\frac{1}{3}$[/tex] as [tex]$n$[/tex] increases by 1, we can reverse the process. To find [tex]$f(1)$[/tex], we first find [tex]$f(2)$[/tex] using the reverse recurrence relation:
[tex]$$
f(3) = \frac{1}{3} f(2) \quad \Longrightarrow \quad f(2) = 3 \cdot f(3) = 3 \cdot 9 = 27.
$$[/tex]
Next, we find [tex]$f(1)$[/tex] similarly:
[tex]$$
f(2) = \frac{1}{3} f(1) \quad \Longrightarrow \quad f(1) = 3 \cdot f(2) = 3 \cdot 27 = 81.
$$[/tex]
Thus, the answer is
[tex]$$
\boxed{81}.
$$[/tex]
