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][tex]$f(1)$[/tex][/tex]?

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

Answer :

Sure, let's solve the problem step by step!

We are given a sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. Also, we know that [tex]\( f(3) = 9 \)[/tex].

We need to find the value of [tex]\( f(1) \)[/tex].

1. Start with [tex]\( f(3) \)[/tex]:
It's given that [tex]\( f(3) = 9 \)[/tex].

2. Find [tex]\( f(2) \)[/tex]:
Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we rearrange to find [tex]\( f(n) \)[/tex] in terms of [tex]\( f(n+1) \)[/tex]:
[tex]\[
f(n) = 3 \times f(n+1)
\][/tex]
Therefore,
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]

3. Find [tex]\( f(1) \)[/tex]:
Use the same rearranged formula:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]

So, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].

Therefore, the correct answer is [tex]\( 81 \)[/tex].