Answer :
We are given the recursive relation
$$
f(n+1)=\frac{1}{3} f(n)
$$
and the value
$$
f(3)=9.
$$
Our goal is to find $f(1)$.
**Step 1.** Use the recursion for $n=2$:
$$
f(3)=\frac{1}{3} f(2).
$$
We substitute $f(3)=9$ to get:
$$
9=\frac{1}{3} f(2).
$$
Multiplying both sides by $3$ gives:
$$
f(2)=3 \times 9=27.
$$
**Step 2.** Now use the recursion for $n=1$:
$$
f(2)=\frac{1}{3} f(1).
$$
Substitute $f(2)=27$:
$$
27=\frac{1}{3} f(1).
$$
Multiplying both sides by $3$ gives:
$$
f(1)=3 \times 27=81.
$$
Thus, the value of $f(1)$ is $\boxed{81}$.
$$
f(n+1)=\frac{1}{3} f(n)
$$
and the value
$$
f(3)=9.
$$
Our goal is to find $f(1)$.
**Step 1.** Use the recursion for $n=2$:
$$
f(3)=\frac{1}{3} f(2).
$$
We substitute $f(3)=9$ to get:
$$
9=\frac{1}{3} f(2).
$$
Multiplying both sides by $3$ gives:
$$
f(2)=3 \times 9=27.
$$
**Step 2.** Now use the recursion for $n=1$:
$$
f(2)=\frac{1}{3} f(1).
$$
Substitute $f(2)=27$:
$$
27=\frac{1}{3} f(1).
$$
Multiplying both sides by $3$ gives:
$$
f(1)=3 \times 27=81.
$$
Thus, the value of $f(1)$ is $\boxed{81}$.
