Answer :
To find [tex]\( f(1) \)[/tex], we will use the recursive relationship given by the function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and the information that [tex]\( f(3) = 9 \)[/tex].
1. Identify the Recursive Relationship:
- The sequence follows the rule [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that to find the next term in the sequence, we divide the current term by 3.
2. Find [tex]\( f(2) \)[/tex]:
- We are given that [tex]\( f(3) = 9 \)[/tex].
- Using the relationship, we need to find the previous term, [tex]\( f(2) \)[/tex]. To do this, we reverse the operation by multiplying by 3:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
- Now that we have [tex]\( f(2) = 27 \)[/tex], we use the same method to find [tex]\( f(1) \)[/tex] by multiplying [tex]\( f(2) \)[/tex] by 3:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
1. Identify the Recursive Relationship:
- The sequence follows the rule [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that to find the next term in the sequence, we divide the current term by 3.
2. Find [tex]\( f(2) \)[/tex]:
- We are given that [tex]\( f(3) = 9 \)[/tex].
- Using the relationship, we need to find the previous term, [tex]\( f(2) \)[/tex]. To do this, we reverse the operation by multiplying by 3:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
- Now that we have [tex]\( f(2) = 27 \)[/tex], we use the same method to find [tex]\( f(1) \)[/tex] by multiplying [tex]\( f(2) \)[/tex] by 3:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
