Answer :
To solve the problem, we need to find the value of [tex]\( f(1) \)[/tex] given the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and that [tex]\( f(3) = 9 \)[/tex].
Let's go through the steps:
1. Understanding the Recursive Relationship:
- The function defines that each term in the sequence is obtained by multiplying the previous term by [tex]\(\frac{1}{3}\)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} \times f(2) \)[/tex] and we're given that [tex]\( f(3) = 9 \)[/tex], we can write:
[tex]\[
9 = \frac{1}{3} \times f(2)
\][/tex]
- Solving for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Similarly, the recursive relationship gives [tex]\( f(2) = \frac{1}{3} \times f(1) \)[/tex].
- Substituting the value we found for [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} \times f(1)
\][/tex]
- Solving 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].
Let's go through the steps:
1. Understanding the Recursive Relationship:
- The function defines that each term in the sequence is obtained by multiplying the previous term by [tex]\(\frac{1}{3}\)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} \times f(2) \)[/tex] and we're given that [tex]\( f(3) = 9 \)[/tex], we can write:
[tex]\[
9 = \frac{1}{3} \times f(2)
\][/tex]
- Solving for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Similarly, the recursive relationship gives [tex]\( f(2) = \frac{1}{3} \times f(1) \)[/tex].
- Substituting the value we found for [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} \times f(1)
\][/tex]
- Solving 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].
