Answer :
To solve this problem, we need to work backward through the recursive sequence using the information provided. We know that the function is defined by:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we are given that [tex]\( f(3) = 9 \)[/tex]. Our goal is to find [tex]\( f(1) \)[/tex].
Let's go step by step:
1. Find [tex]\( f(2) \)[/tex]:
According to the recursive definition, to find [tex]\( f(n) \)[/tex] from [tex]\( f(n+1) \)[/tex], we can rearrange the formula:
[tex]\[ f(n) = 3 \times f(n+1) \][/tex]
We know [tex]\( f(3) = 9 \)[/tex], so let's calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 3 \times f(3) = 3 \times 9 = 27 \][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Similarly, using the same method, we need to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[ f(1) = 3 \times f(2) = 3 \times 27 = 81 \][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we are given that [tex]\( f(3) = 9 \)[/tex]. Our goal is to find [tex]\( f(1) \)[/tex].
Let's go step by step:
1. Find [tex]\( f(2) \)[/tex]:
According to the recursive definition, to find [tex]\( f(n) \)[/tex] from [tex]\( f(n+1) \)[/tex], we can rearrange the formula:
[tex]\[ f(n) = 3 \times f(n+1) \][/tex]
We know [tex]\( f(3) = 9 \)[/tex], so let's calculate [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 3 \times f(3) = 3 \times 9 = 27 \][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Similarly, using the same method, we need to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[ f(1) = 3 \times f(2) = 3 \times 27 = 81 \][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
