High School

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A sequence is defined by the recursive function [tex]f(n+1) = \frac{1}{3} f(n)[/tex]. If [tex]f(3) = 9[/tex], what is [tex]f(1)[/tex]?

A. 1
B. 3
C. 27
D. 81

Answer :

To solve the problem, we need to find the value of [tex]\( f(1) \)[/tex] given the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and that [tex]\( f(3) = 9 \)[/tex].

Let's go through the steps:

1. Understanding the Recursive Relationship:
- The function defines that each term in the sequence is obtained by multiplying the previous term by [tex]\(\frac{1}{3}\)[/tex].

2. Finding [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} \times f(2) \)[/tex] and we're given that [tex]\( f(3) = 9 \)[/tex], we can write:
[tex]\[
9 = \frac{1}{3} \times f(2)
\][/tex]
- Solving for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

3. Finding [tex]\( f(1) \)[/tex]:
- Similarly, the recursive relationship gives [tex]\( f(2) = \frac{1}{3} \times f(1) \)[/tex].
- Substituting the value we found for [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} \times f(1)
\][/tex]
- Solving for [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( \boxed{81} \)[/tex].

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