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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 :

Let's solve this problem step-by-step.

We are given a sequence defined by a recursive function:

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

This means each term in the sequence is one third of the previous term.

We also know that:

[tex]\[ f(3) = 9 \][/tex]

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

To solve this, let's backtrack from [tex]\( f(3) \)[/tex] to [tex]\( f(1) \)[/tex]:

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

Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can rearrange this to find [tex]\( f(2) \)[/tex]:

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

Given [tex]\( f(3) = 9 \)[/tex], we substitute:

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

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

Now, use the same approach for [tex]\( f(2) \)[/tex] and [tex]\( f(1) \)[/tex]:

Since [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], rearrange to find [tex]\( f(1) \)[/tex]:

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

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

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

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

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