Answer :
To solve the problem, we need to find the value of [tex]\( f(1) \)[/tex] given the sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and the value [tex]\( f(3) = 9 \)[/tex].
Let's work backwards from the given information:
1. Start with [tex]\( f(3) \)[/tex]:
We know [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
According to the recursive rule:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute the known value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Again, using the recursive rule:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Let's work backwards from the given information:
1. Start with [tex]\( f(3) \)[/tex]:
We know [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
According to the recursive rule:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute the known value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Again, using the recursive rule:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
