College

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 for [tex]\( f(1) \)[/tex] in the given sequence, we need to work backwards from the information provided.

The sequence is defined by the recursive formula:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we know:
[tex]\[ f(3) = 9 \][/tex]

Let's find [tex]\( f(2) \)[/tex]:

From the recursive formula, we have:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]

Since [tex]\( f(3) = 9 \)[/tex], we can substitute this into the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]

To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]

Next, we find [tex]\( f(1) \)[/tex]:

Using the recursive formula again, we know:
[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]

To solve 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].