Answer :
Let's solve the problem step by step:
We have a sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. We know that [tex]\( f(3) = 9 \)[/tex] and we need to find [tex]\( f(1) \)[/tex].
1. Find [tex]\( f(2) \)[/tex]:
According to the recursive formula, [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
We know [tex]\( f(3) = 9 \)[/tex], so we can set up the equation:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Similarly, using the recursive formula [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex],
we have [tex]\( f(2) = 27 \)[/tex], so:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To solve for [tex]\( f(1) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
We have a sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. We know that [tex]\( f(3) = 9 \)[/tex] and we need to find [tex]\( f(1) \)[/tex].
1. Find [tex]\( f(2) \)[/tex]:
According to the recursive formula, [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
We know [tex]\( f(3) = 9 \)[/tex], so we can set up the equation:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Similarly, using the recursive formula [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex],
we have [tex]\( f(2) = 27 \)[/tex], so:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To solve for [tex]\( f(1) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
