Answer :
We are given the recursive relation
[tex]$$
f(n+1) = \frac{1}{3} f(n)
$$[/tex]
and the value
[tex]$$
f(3)=9.
$$[/tex]
Since the recursion relates a term to its previous term, we can rearrange the relation to express the previous term in terms of the next term:
[tex]$$
f(n) = 3 \cdot f(n+1).
$$[/tex]
To find [tex]$f(1)$[/tex], we first find [tex]$f(2)$[/tex] by setting [tex]$n = 2$[/tex]:
[tex]$$
f(2) = 3 \cdot f(3) = 3 \cdot 9 = 27.
$$[/tex]
Next, we use [tex]$f(2)$[/tex] to calculate [tex]$f(1)$[/tex] by setting [tex]$n = 1$[/tex]:
[tex]$$
f(1) = 3 \cdot f(2) = 3 \cdot 27 = 81.
$$[/tex]
Thus, the value of [tex]$f(1)$[/tex] 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 recursion relates a term to its previous term, we can rearrange the relation to express the previous term in terms of the next term:
[tex]$$
f(n) = 3 \cdot f(n+1).
$$[/tex]
To find [tex]$f(1)$[/tex], we first find [tex]$f(2)$[/tex] by setting [tex]$n = 2$[/tex]:
[tex]$$
f(2) = 3 \cdot f(3) = 3 \cdot 9 = 27.
$$[/tex]
Next, we use [tex]$f(2)$[/tex] to calculate [tex]$f(1)$[/tex] by setting [tex]$n = 1$[/tex]:
[tex]$$
f(1) = 3 \cdot f(2) = 3 \cdot 27 = 81.
$$[/tex]
Thus, the value of [tex]$f(1)$[/tex] is [tex]$\boxed{81}$[/tex].
