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]
Step 1. Find [tex]\( f(2) \)[/tex]:
Since the recursive relation is
[tex]$$
f(3)=\frac{1}{3}f(2),
$$[/tex]
we can solve for [tex]\( f(2) \)[/tex] by multiplying both sides by 3:
[tex]$$
f(2)=3f(3)=3 \times 9=27.
$$[/tex]
Step 2. Find [tex]\( f(1) \)[/tex]:
Using the recursive formula again for [tex]\( n=1 \)[/tex], we have:
[tex]$$
f(2)=\frac{1}{3}f(1).
$$[/tex]
Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]$$
f(1)=3f(2)=3 \times 27=81.
$$[/tex]
Thus, the final answer is
[tex]$$
\boxed{81}.
$$[/tex]
[tex]$$
f(n+1)=\frac{1}{3}f(n)
$$[/tex]
and the value
[tex]$$
f(3)=9.
$$[/tex]
Step 1. Find [tex]\( f(2) \)[/tex]:
Since the recursive relation is
[tex]$$
f(3)=\frac{1}{3}f(2),
$$[/tex]
we can solve for [tex]\( f(2) \)[/tex] by multiplying both sides by 3:
[tex]$$
f(2)=3f(3)=3 \times 9=27.
$$[/tex]
Step 2. Find [tex]\( f(1) \)[/tex]:
Using the recursive formula again for [tex]\( n=1 \)[/tex], we have:
[tex]$$
f(2)=\frac{1}{3}f(1).
$$[/tex]
Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]$$
f(1)=3f(2)=3 \times 27=81.
$$[/tex]
Thus, the final answer is
[tex]$$
\boxed{81}.
$$[/tex]
