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Answer :
To solve this problem, we need to understand the sequence is defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], and we know that [tex]\( f(3) = 9 \)[/tex]. We are tasked with finding the value of [tex]\( f(1) \)[/tex].
Let's work through this step-by-step:
1. Understand the Recursive Relation: The function given is a recursive relationship where the next term [tex]\( f(n+1) \)[/tex] is one-third of the previous term [tex]\( f(n) \)[/tex].
2. Given Information: We know [tex]\( f(3) = 9 \)[/tex].
3. Finding [tex]\( f(2) \)[/tex]: To go backwards from [tex]\( f(3) \)[/tex] to find [tex]\( f(2) \)[/tex], we need to reverse the recursive formula. If [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], then it follows that [tex]\( f(n) = 3 \times f(n+1) \)[/tex].
- Since [tex]\( f(3) = 9 \)[/tex], we can find [tex]\( f(2) \)[/tex] by multiplying 9 by 3:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
4. Finding [tex]\( f(1) \)[/tex]: Similarly, use the same process to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(2) = 27 \)[/tex], we can find [tex]\( f(1) \)[/tex] by multiplying 27 by 3:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Let's work through this step-by-step:
1. Understand the Recursive Relation: The function given is a recursive relationship where the next term [tex]\( f(n+1) \)[/tex] is one-third of the previous term [tex]\( f(n) \)[/tex].
2. Given Information: We know [tex]\( f(3) = 9 \)[/tex].
3. Finding [tex]\( f(2) \)[/tex]: To go backwards from [tex]\( f(3) \)[/tex] to find [tex]\( f(2) \)[/tex], we need to reverse the recursive formula. If [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], then it follows that [tex]\( f(n) = 3 \times f(n+1) \)[/tex].
- Since [tex]\( f(3) = 9 \)[/tex], we can find [tex]\( f(2) \)[/tex] by multiplying 9 by 3:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
4. Finding [tex]\( f(1) \)[/tex]: Similarly, use the same process to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(2) = 27 \)[/tex], we can find [tex]\( f(1) \)[/tex] by multiplying 27 by 3:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
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