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 are given a sequence defined by the recursive function [tex]\(f(n+1) = \frac{1}{3} f(n)\)[/tex], and a specific value [tex]\(f(3) = 9\)[/tex]. We need to find the value of [tex]\(f(1)\)[/tex].

Let's break this down step-by-step:

1. Identify the Relationship:
The recursive relation [tex]\(f(n+1) = \frac{1}{3} f(n)\)[/tex] signifies that each term in the sequence is one-third of the previous term.

2. Work Backwards from [tex]\(f(3)\)[/tex]:
We know:
[tex]\[
f(3) = \frac{1}{3} \times f(2)
\][/tex]
Given [tex]\(f(3) = 9\)[/tex], we can find [tex]\(f(2)\)[/tex] by rearranging the equation:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]

3. Find [tex]\(f(1)\)[/tex]:
Now that we have [tex]\(f(2)\)[/tex], use the recursive relation to find [tex]\(f(1)\)[/tex]:
[tex]\[
f(2) = \frac{1}{3} \times f(1)
\][/tex]
Substituting [tex]\(f(2) = 27\)[/tex], we rearrange to find [tex]\(f(1)\)[/tex]:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]

Thus, the value of [tex]\(f(1)\)[/tex] is [tex]\(\boxed{81}\)[/tex].

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