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Answer :
To solve the problem, we need to determine the value of [tex]\( f(1) \)[/tex] from the recursively defined sequence where [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and it is given that [tex]\( f(3) = 9 \)[/tex].
1. Understanding the Sequence:
The sequence is defined recursively. This means each term in the sequence is derived from the previous term according to a specific rule. In this case, each term is one-third of the previous term.
2. Find [tex]\( f(2) \)[/tex]:
We know that [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. To find [tex]\( f(2) \)[/tex], we work backwards from [tex]\( f(3) \)[/tex].
From the recursive formula, we can express:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
We know [tex]\( f(3) = 9 \)[/tex]. So,
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
By solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Now that we know [tex]\( f(2) = 27 \)[/tex], we can use the formula once again to find [tex]\( f(1) \)[/tex].
According to the formula:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the known value of [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex] involves multiplying both sides by 3:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
1. Understanding the Sequence:
The sequence is defined recursively. This means each term in the sequence is derived from the previous term according to a specific rule. In this case, each term is one-third of the previous term.
2. Find [tex]\( f(2) \)[/tex]:
We know that [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. To find [tex]\( f(2) \)[/tex], we work backwards from [tex]\( f(3) \)[/tex].
From the recursive formula, we can express:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
We know [tex]\( f(3) = 9 \)[/tex]. So,
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
By solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Now that we know [tex]\( f(2) = 27 \)[/tex], we can use the formula once again to find [tex]\( f(1) \)[/tex].
According to the formula:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the known value of [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex] involves multiplying both sides by 3:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
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