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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 :

Sure, let's break down the problem step by step to find [tex]\( f(1) \)[/tex] using the given recursive function.

The recursive function is:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]

We are given:
[tex]\[ f(3) = 9 \][/tex]

We need to find [tex]\( f(1) \)[/tex].

1. First, let's find [tex]\( f(2) \)[/tex]. According to the recursive function:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Since [tex]\( f(3) = 9 \)[/tex]:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]

2. Next, let's find [tex]\( f(1) \)[/tex]. Again, using the recursive function:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Since [tex]\( f(2) = 27 \)[/tex]:
[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]

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

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