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Answer :
To solve this problem, we need to understand the recursive sequence defined by the function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that each term is one-third of the previous term. We are given that [tex]\( f(3) = 9 \)[/tex], and we want to find [tex]\( f(1) \)[/tex].
Let's work backwards from the provided information:
1. Given: [tex]\( f(3) = 9 \)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can rearrange this to find [tex]\( f(n) \)[/tex] if we know [tex]\( f(n+1) \)[/tex]:
[tex]\[
f(n) = 3 \times f(n+1)
\][/tex]
So, to find [tex]\( f(2) \)[/tex], we calculate:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
Similarly, to find [tex]\( f(1) \)[/tex], use the same rearranged expression:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Thus, [tex]\( f(1) = 81 \)[/tex].
Therefore, the initial term of the sequence, [tex]\( f(1) \)[/tex], is [tex]\( \boxed{81} \)[/tex].
Let's work backwards from the provided information:
1. Given: [tex]\( f(3) = 9 \)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can rearrange this to find [tex]\( f(n) \)[/tex] if we know [tex]\( f(n+1) \)[/tex]:
[tex]\[
f(n) = 3 \times f(n+1)
\][/tex]
So, to find [tex]\( f(2) \)[/tex], we calculate:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
Similarly, to find [tex]\( f(1) \)[/tex], use the same rearranged expression:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Thus, [tex]\( f(1) = 81 \)[/tex].
Therefore, the initial term of the sequence, [tex]\( f(1) \)[/tex], is [tex]\( \boxed{81} \)[/tex].
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