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Answer :
To solve the problem, we need to find [tex]\( f(1) \)[/tex] given that the sequence is defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], and [tex]\( f(3) = 9 \)[/tex].
Here's how we can solve it step-by-step:
1. Understand the Recursive Formula:
The formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] tells us that to get the next term in the sequence, we multiply the current term by [tex]\(\frac{1}{3}\)[/tex].
2. Reverse the Process:
If we know a term in the sequence and want to find the previous one, we need to do the opposite operation. So, to find [tex]\( f(n) \)[/tex] from [tex]\( f(n+1) \)[/tex], we multiply by 3.
3. Use the Given Information:
We know [tex]\( f(3) = 9 \)[/tex].
4. Find [tex]\( f(2) \)[/tex]:
To find [tex]\( f(2) \)[/tex], we reverse the recursive step from [tex]\( f(3) \)[/tex]:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
5. Find [tex]\( f(1) \)[/tex]:
Now, use the same method to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is 81.
Here's how we can solve it step-by-step:
1. Understand the Recursive Formula:
The formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] tells us that to get the next term in the sequence, we multiply the current term by [tex]\(\frac{1}{3}\)[/tex].
2. Reverse the Process:
If we know a term in the sequence and want to find the previous one, we need to do the opposite operation. So, to find [tex]\( f(n) \)[/tex] from [tex]\( f(n+1) \)[/tex], we multiply by 3.
3. Use the Given Information:
We know [tex]\( f(3) = 9 \)[/tex].
4. Find [tex]\( f(2) \)[/tex]:
To find [tex]\( f(2) \)[/tex], we reverse the recursive step from [tex]\( f(3) \)[/tex]:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
5. Find [tex]\( f(1) \)[/tex]:
Now, use the same method to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is 81.
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