Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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

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

Here's how you can solve it step by step:

1. Understand the Recursive Function:
- The function tells us how to get from one term to the next in the sequence. Specifically, each term is one-third of the previous term.

2. Work Backwards from the Given Information:
- We are given [tex]$ f(3) = 9 $[/tex]. Our goal is to find [tex]$ f(2) $[/tex] and then [tex]$ f(1) $[/tex].

3. Find [tex]$ f(2) $[/tex]:
- According to the recursive function, [tex]$ f(3) = \frac{1}{3} f(2) $[/tex].
- Since [tex]$ f(3) = 9 $[/tex], we can set up the equation:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
- Multiply both sides by 3 to solve for [tex]$ f(2) $[/tex]:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

4. Find [tex]$ f(1) $[/tex]:
- Now use the same recursive formula to find [tex]$ f(1) $[/tex]. According to the formula, [tex]$ f(2) = \frac{1}{3} f(1) $[/tex].
- We already found [tex]$ f(2) = 27 $[/tex], so:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
- Multiply both sides by 3 to solve for [tex]$ f(1) $[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

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