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]f(1)[/tex]?

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

Answer :

To solve this problem, we need to determine the value of [tex]\( f(1) \)[/tex] given the recursive function and the value of [tex]\( f(3) \)[/tex].

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

And we know that:
[tex]\[ f(3) = 9 \][/tex]

To find [tex]\( f(1) \)[/tex], we can work backwards using the recursive relationship. Here are the steps:

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

According to the recursive formula:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]

Since we know [tex]\( f(3) = 9 \)[/tex], we can set up the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]

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

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

Now that we know [tex]\( f(2) = 27 \)[/tex], use the recursive formula again to find [tex]\( f(1) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]

Substituting the known value of [tex]\( f(2) \)[/tex]:
[tex]\[ 27 = \frac{1}{3} f(1) \][/tex]

Solving 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 81.

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