Answer :
To solve the problem, let's work through the sequence using the recursive function provided:
We are given that the sequence is defined by the function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that each term of the sequence is one-third of the previous term.
We also know that [tex]\( f(3) = 9 \)[/tex] and we need to find the initial term of the sequence, [tex]\( f(0) \)[/tex].
Let's work backwards from the known term:
1. Since [tex]\( f(3) = 9 \)[/tex], we can find [tex]\( f(2) \)[/tex] by recognizing that:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex] gives:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Now, using [tex]\( f(2) = 27 \)[/tex], we find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex] gives:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
3. Finally, using [tex]\( f(1) = 81 \)[/tex], we find [tex]\( f(0) \)[/tex]:
[tex]\[
f(1) = \frac{1}{3} f(0)
\][/tex]
Solving for [tex]\( f(0) \)[/tex] gives:
[tex]\[
f(0) = 81 \times 3 = 243
\][/tex]
Therefore, the initial term [tex]\( f(0) \)[/tex] of the sequence is [tex]\(\boxed{243}\)[/tex].
We are given that the sequence is defined by the function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that each term of the sequence is one-third of the previous term.
We also know that [tex]\( f(3) = 9 \)[/tex] and we need to find the initial term of the sequence, [tex]\( f(0) \)[/tex].
Let's work backwards from the known term:
1. Since [tex]\( f(3) = 9 \)[/tex], we can find [tex]\( f(2) \)[/tex] by recognizing that:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex] gives:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Now, using [tex]\( f(2) = 27 \)[/tex], we find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex] gives:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
3. Finally, using [tex]\( f(1) = 81 \)[/tex], we find [tex]\( f(0) \)[/tex]:
[tex]\[
f(1) = \frac{1}{3} f(0)
\][/tex]
Solving for [tex]\( f(0) \)[/tex] gives:
[tex]\[
f(0) = 81 \times 3 = 243
\][/tex]
Therefore, the initial term [tex]\( f(0) \)[/tex] of the sequence is [tex]\(\boxed{243}\)[/tex].
