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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 solve this problem, we need to find the value of [tex]\( f(1) \)[/tex] in a sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], given that [tex]\( f(3) = 9 \)[/tex].

Let's break it down step-by-step:

1. Find [tex]\( f(2) \)[/tex]:

We are given that [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. Using this recursive relationship, we can write:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]

Since [tex]\( f(3) = 9 \)[/tex], we substitute this value into the equation:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]

To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

2. Find [tex]\( f(1) \)[/tex]:

Now, using the same recursive formula, we have:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]

Substitute the value of [tex]\( f(2) \)[/tex] that we found:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]

To solve for [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

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

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