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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]f(1)[/tex]?

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

Answer :

To solve the problem, we need to determine the value of [tex]\( f(1) \)[/tex] given the recursive relationship and a specific term in the sequence. Let's break it down step-by-step.

1. Understand the Recursive Relationship:
The sequence is defined recursively as [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term is one-third of the previous term.

2. Given Information:
We know that [tex]\( f(3) = 9 \)[/tex].

3. Find [tex]\( f(2) \)[/tex]:
According to the recursive relationship:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute the known value [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 27
\][/tex]

4. Find [tex]\( f(1) \)[/tex]:
Use the recursive relationship again:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the known value [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 81
\][/tex]

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

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