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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 of finding [tex]\( f(1) \)[/tex] for the given recursive sequence, we start by understanding the sequence definition: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].

Given that [tex]\( f(3) = 9 \)[/tex], let's work backwards to find [tex]\( f(1) \)[/tex]:

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

According to the recursive formula, if [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], then to go backwards we can multiply by 3. So, we have:

[tex]\[
f(2) = 3 \times f(3)
\][/tex]

Substitute the given value of [tex]\( f(3) = 9 \)[/tex]:

[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]

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

Similarly, if [tex]\( f(2) = 3 \times f(3) \)[/tex], then to find [tex]\( f(1) \)[/tex], we use:

[tex]\[
f(1) = 3 \times f(2)
\][/tex]

Substitute the value of [tex]\( f(2) = 27 \)[/tex]:

[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]

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

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