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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, let's go through the sequence using the given information:

1. We are given the recursive function for the sequence: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
2. We also know that [tex]\( f(3) = 9 \)[/tex] and need to find [tex]\( f(1) \)[/tex].

Let's work backwards from what we know:

Step 1: From the definition [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(3) \)[/tex]:

[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]

Since [tex]\( f(3) = 9 \)[/tex], we substitute this into the equation:

[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]

To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:

[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]

Step 2: Now, using the same recursive relation, we can express [tex]\( f(1) \)[/tex] in terms of [tex]\( f(2) \)[/tex]:

[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]

Substitute the value of [tex]\( f(2) = 27 \)[/tex] into the equation:

[tex]\[ 27 = \frac{1}{3} f(1) \][/tex]

To solve for [tex]\( f(1) \)[/tex], multiply both sides by 3:

[tex]\[ f(1) = 27 \times 3 = 81 \][/tex]

Therefore, [tex]\( f(1) = 81 \)[/tex].

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