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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 the problem, we need to determine the initial term of the sequence, [tex]\( f(1) \)[/tex], using the recursive function and the given value [tex]\( f(3) = 9 \)[/tex].

1. Understanding the sequence:
The sequence is defined by the recursive formula:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
This means each term is one-third of the previous term.

2. Working backwards from [tex]\( f(3) = 9 \)[/tex]:
We know:
[tex]\[
f(3) = 9
\][/tex]
Using the recursive formula, we can express [tex]\( f(3) \)[/tex] in terms of [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substituting the given value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

3. Determining [tex]\( f(1) \)[/tex]:
Similarly, using the recursive formula for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the value we found for [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].

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