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Answer :
To solve the problem, let's work through finding [tex]\( f(1) \)[/tex] based on the recursive sequence defined by the function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and given that [tex]\( f(3) = 9 \)[/tex].
Here’s a step-by-step breakdown:
1. Understand the recursive relationship:
The formula tells us that each term in the sequence is one third of the previous term. So, if we know a term, we can use it to find the previous term by multiplying by 3.
2. Find [tex]\( f(2) \)[/tex]:
We are given [tex]\( f(3) = 9 \)[/tex]. We need to find [tex]\( f(2) \)[/tex], which is the term before [tex]\( f(3) \)[/tex].
[tex]\[
f(3) = \frac{1}{3} f(2) \implies f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Now that we have [tex]\( f(2) = 27 \)[/tex], we can find [tex]\( f(1) \)[/tex], which is the term before [tex]\( f(2) \)[/tex].
[tex]\[
f(2) = \frac{1}{3} f(1) \implies f(1) = 27 \times 3 = 81
\][/tex]
Consequently, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
Here’s a step-by-step breakdown:
1. Understand the recursive relationship:
The formula tells us that each term in the sequence is one third of the previous term. So, if we know a term, we can use it to find the previous term by multiplying by 3.
2. Find [tex]\( f(2) \)[/tex]:
We are given [tex]\( f(3) = 9 \)[/tex]. We need to find [tex]\( f(2) \)[/tex], which is the term before [tex]\( f(3) \)[/tex].
[tex]\[
f(3) = \frac{1}{3} f(2) \implies f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Now that we have [tex]\( f(2) = 27 \)[/tex], we can find [tex]\( f(1) \)[/tex], which is the term before [tex]\( f(2) \)[/tex].
[tex]\[
f(2) = \frac{1}{3} f(1) \implies f(1) = 27 \times 3 = 81
\][/tex]
Consequently, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
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