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Answer :
Sure, let's solve the problem step by step!
We are given a sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. Also, we know that [tex]\( f(3) = 9 \)[/tex].
We need to find the value of [tex]\( f(1) \)[/tex].
1. Start with [tex]\( f(3) \)[/tex]:
It's given that [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we rearrange to find [tex]\( f(n) \)[/tex] in terms of [tex]\( f(n+1) \)[/tex]:
[tex]\[
f(n) = 3 \times f(n+1)
\][/tex]
Therefore,
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Use the same rearranged formula:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Therefore, the correct answer is [tex]\( 81 \)[/tex].
We are given a sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. Also, we know that [tex]\( f(3) = 9 \)[/tex].
We need to find the value of [tex]\( f(1) \)[/tex].
1. Start with [tex]\( f(3) \)[/tex]:
It's given that [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we rearrange to find [tex]\( f(n) \)[/tex] in terms of [tex]\( f(n+1) \)[/tex]:
[tex]\[
f(n) = 3 \times f(n+1)
\][/tex]
Therefore,
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Use the same rearranged formula:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Therefore, the correct answer is [tex]\( 81 \)[/tex].
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