College

We appreciate your visit to 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. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

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 [tex]\( f(1) \)[/tex] given the recursive function and the value [tex]\( f(3) = 9 \)[/tex].

1. Understanding the Recurrence:
The sequence is defined by the recursive function:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
This means that each term in the sequence is one-third of the previous term.

2. Work Backwards from [tex]\( f(3) \)[/tex]:

Since we're told that [tex]\( f(3) = 9 \)[/tex], let's use the recursion to find earlier terms:

- Finding [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2) \Rightarrow 9 = \frac{1}{3} f(2)
\][/tex]
Solve for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

- Finding [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1) \Rightarrow 27 = \frac{1}{3} f(1)
\][/tex]
Solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

3. Conclusion:
Thus, the value of [tex]\( f(1) \)[/tex] is 81.

Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada