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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 are given a recursive function for a sequence: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].

We also know that [tex]\( f(3) = 9 \)[/tex].

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

Step 1: Express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(3) \)[/tex]

From the recursive relationship, we have:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]

To find [tex]\( f(2) \)[/tex], we rearrange this to:
[tex]\[ f(2) = 3 \times f(3) \][/tex]

Given [tex]\( f(3) = 9 \)[/tex], substitute this value in:
[tex]\[ f(2) = 3 \times 9 = 27 \][/tex]

Step 2: Express [tex]\( f(1) \)[/tex] in terms of [tex]\( f(2) \)[/tex]

Using the recursive relationship again:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]

To find [tex]\( f(1) \)[/tex], rearrange this to:
[tex]\[ f(1) = 3 \times f(2) \][/tex]

Now substitute [tex]\( f(2) = 27 \)[/tex] in:
[tex]\[ f(1) = 3 \times 27 = 81 \][/tex]

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

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