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 :

Sure! Let's solve the problem step by step:

We have a sequence described by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], and we know that [tex]\( f(3) = 9 \)[/tex]. We need to find [tex]\( f(1) \)[/tex].

Step 1: Determine [tex]\( f(2) \)[/tex] from [tex]\( f(3) \)[/tex].

Given that [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can write:

[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]

We know [tex]\( f(3) = 9 \)[/tex], so:

[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]

To find [tex]\( f(2) \)[/tex], multiply both sides of the equation by 3:

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

Step 2: Determine [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex].

Similarly, using the recursive definition, we can write:

[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]

We found that [tex]\( f(2) = 27 \)[/tex], so:

[tex]\[ 27 = \frac{1}{3} f(1) \][/tex]

To find [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 81.

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