Answer :
Sure! Let's solve this step-by-step using the given recursive function.
We are given the following:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
and
[tex]\[ f(3) = 9 \][/tex]
We want to find [tex]\( f(1) \)[/tex].
First, let's work our way backward from [tex]\( f(3) \)[/tex] to [tex]\( f(1) \)[/tex]:
1. Find [tex]\( f(2) \)[/tex] from [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
We know [tex]\( f(3) = 9 \)[/tex], so:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 \][/tex]
[tex]\[ f(2) = 27 \][/tex]
2. Find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
We now know [tex]\( f(2) = 27 \)[/tex], so:
[tex]\[ 27 = \frac{1}{3} f(1) \][/tex]
To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[ f(1) = 27 \times 3 \][/tex]
[tex]\[ f(1) = 81 \][/tex]
So, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Therefore, the correct answer is:
[tex]\[ 81 \][/tex]
We are given the following:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
and
[tex]\[ f(3) = 9 \][/tex]
We want to find [tex]\( f(1) \)[/tex].
First, let's work our way backward from [tex]\( f(3) \)[/tex] to [tex]\( f(1) \)[/tex]:
1. Find [tex]\( f(2) \)[/tex] from [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
We know [tex]\( f(3) = 9 \)[/tex], so:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 \][/tex]
[tex]\[ f(2) = 27 \][/tex]
2. Find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
We now know [tex]\( f(2) = 27 \)[/tex], so:
[tex]\[ 27 = \frac{1}{3} f(1) \][/tex]
To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[ f(1) = 27 \times 3 \][/tex]
[tex]\[ f(1) = 81 \][/tex]
So, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Therefore, the correct answer is:
[tex]\[ 81 \][/tex]
