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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 the problem step by step:

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

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

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

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

To solve for [tex]\( f(2) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

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

Similarly, using the recursive formula [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex],

we have [tex]\( f(2) = 27 \)[/tex], so:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]

To solve for [tex]\( f(1) \)[/tex], multiply both sides of the equation 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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