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Answer :
To solve this problem, we need to find the initial term [tex]\( f(1) \)[/tex] of the sequence, given that [tex]\( f(3) = 9 \)[/tex] and the sequence is defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
Let's break it down step by step:
1. Understand the recursive relationship: The formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] indicates that each term in the sequence is one-third the value of the previous term.
2. Work backwards from the known value: We know [tex]\( f(3) = 9 \)[/tex]. According to the recursive relationship, [tex]\( f(2) \)[/tex] would be the term that comes just before [tex]\( f(3) \)[/tex]. We can set up the equation for [tex]\( f(3) \)[/tex] as follows:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Plug in the known value [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
Multiplying both sides by 3 to solve for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex] using the same method: Now that we have [tex]\( f(2) = 27 \)[/tex], we can use the recursive relationship again to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex] into the equation:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Let's break it down step by step:
1. Understand the recursive relationship: The formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] indicates that each term in the sequence is one-third the value of the previous term.
2. Work backwards from the known value: We know [tex]\( f(3) = 9 \)[/tex]. According to the recursive relationship, [tex]\( f(2) \)[/tex] would be the term that comes just before [tex]\( f(3) \)[/tex]. We can set up the equation for [tex]\( f(3) \)[/tex] as follows:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Plug in the known value [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
Multiplying both sides by 3 to solve for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex] using the same method: Now that we have [tex]\( f(2) = 27 \)[/tex], we can use the recursive relationship again to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex] into the equation:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
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