Answer :
To solve this problem, we need to determine the value of [tex]\( f(1) \)[/tex] given the recursive function and the value of [tex]\( f(3) \)[/tex].
The recursive function given is:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we know that:
[tex]\[ f(3) = 9 \][/tex]
To find [tex]\( f(1) \)[/tex], we can work backwards using the recursive relationship. Here are the steps:
1. Find [tex]\( f(2) \)[/tex]:
According to the recursive formula:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Since we know [tex]\( f(3) = 9 \)[/tex], we can set up the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
Solving for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Now that we know [tex]\( f(2) = 27 \)[/tex], use the recursive formula again to find [tex]\( f(1) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Substituting the known value of [tex]\( f(2) \)[/tex]:
[tex]\[ 27 = \frac{1}{3} f(1) \][/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 81.
The recursive function given is:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we know that:
[tex]\[ f(3) = 9 \][/tex]
To find [tex]\( f(1) \)[/tex], we can work backwards using the recursive relationship. Here are the steps:
1. Find [tex]\( f(2) \)[/tex]:
According to the recursive formula:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Since we know [tex]\( f(3) = 9 \)[/tex], we can set up the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
Solving for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Now that we know [tex]\( f(2) = 27 \)[/tex], use the recursive formula again to find [tex]\( f(1) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Substituting the known value of [tex]\( f(2) \)[/tex]:
[tex]\[ 27 = \frac{1}{3} f(1) \][/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 81.
