High School

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A sequence is defined by the recursive function [tex]$f(n+1)=\frac{1}{3} f(n)$[/tex]. If [tex]$f(3)=9$[/tex], what is [tex][tex]$f(1)$[/tex][/tex]?

A. 1
B. 3
C. 27
D. 81

Answer :

To find [tex]\( f(1) \)[/tex] in the given recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], and knowing that [tex]\( f(3) = 9 \)[/tex], we'll trace back to find the initial value.

1. We start with [tex]\( f(3) = 9 \)[/tex].

2. To find [tex]\( f(2) \)[/tex], we use the recursive function:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substituting the known value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

3. Next, to find [tex]\( f(1) \)[/tex], we again use the recursive function:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substituting the value of [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].

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