Answer:
81
Step-by-step explanation:
[tex]f(n+1)=\frac{1}{3}f(n)[/tex] means a term can be found by multiplying previous term by [tex]\frac{1}{3}[/tex].
So if we want to go backwards to find the previous term from a term that comes after it we need to multiply that by [tex]3[/tex].
This means:
[tex]f(3)=9[/tex] implies [tex]f(2)=3(9)=27[/tex].
[tex]f(2)=27[/tex] implies [tex]f(1)=3(27)=81[/tex].
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Another way to look at it:
[tex]f(n+1)=\frac{1}{3}f(n)[/tex]
Multiply both sides by 3:
[tex]3f(n+1)=f(n)[/tex]
By the symmetric property:
[tex]f(n)=3f(n+1)[/tex]
So if [tex]n=2[/tex]:
[tex]f(2)=3f(2+1)[/tex]
[tex]f(2)=3f(3)[/tex]
[tex]f(2)=3(9)=27[/tex]
If [tex]n=1[/tex]:
[tex]f(1)=3f(1+1)[/tex]
[tex]f(1)=3f(2)[/tex]
[tex]f(1)=3(27)=81[/tex].
