Answer :
To solve the problem, we start by understanding the sequence defined by the 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 are given [tex]\( f(3) = 9 \)[/tex] and need to find [tex]\( f(1) \)[/tex].
Let's work through the steps:
1. Find [tex]\( f(2) \)[/tex] using [tex]\( f(3) \)[/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]
Substituting the given value [tex]\( f(3) = 9 \)[/tex], we have:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex] using [tex]\( f(2) \)[/tex]:
Similarly, using the recursive definition for the next step, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], rearrange to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2)
\][/tex]
Substituting the value of [tex]\( f(2) = 27 \)[/tex], we calculate:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
We are given [tex]\( f(3) = 9 \)[/tex] and need to find [tex]\( f(1) \)[/tex].
Let's work through the steps:
1. Find [tex]\( f(2) \)[/tex] using [tex]\( f(3) \)[/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]
Substituting the given value [tex]\( f(3) = 9 \)[/tex], we have:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex] using [tex]\( f(2) \)[/tex]:
Similarly, using the recursive definition for the next step, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], rearrange to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2)
\][/tex]
Substituting the value of [tex]\( f(2) = 27 \)[/tex], we calculate:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
