Answer :
To find [tex]\( f(1) \)[/tex] for the given recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] starting with [tex]\( f(3) = 9 \)[/tex], we can proceed step-by-step as follows:
1. Determine [tex]\( f(2) \)[/tex]:
Using the recursion [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], substitute [tex]\( n = 2 \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Given that [tex]\( f(3) = 9 \)[/tex], we solve for [tex]\( f(2) \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2) \implies f(2) = 9 \times 3 = 27
\][/tex]
2. Determine [tex]\( f(1) \)[/tex]:
Again, using the recursion [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], substitute [tex]\( n = 1 \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
We found [tex]\( f(2) = 27 \)[/tex], so we solve for [tex]\( f(1) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1) \implies f(1) = 27 \times 3 = 81
\][/tex]
Therefore, [tex]\( f(1) \)[/tex] is [tex]\( \boxed{81} \)[/tex].
1. Determine [tex]\( f(2) \)[/tex]:
Using the recursion [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], substitute [tex]\( n = 2 \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Given that [tex]\( f(3) = 9 \)[/tex], we solve for [tex]\( f(2) \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2) \implies f(2) = 9 \times 3 = 27
\][/tex]
2. Determine [tex]\( f(1) \)[/tex]:
Again, using the recursion [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], substitute [tex]\( n = 1 \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
We found [tex]\( f(2) = 27 \)[/tex], so we solve for [tex]\( f(1) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1) \implies f(1) = 27 \times 3 = 81
\][/tex]
Therefore, [tex]\( f(1) \)[/tex] is [tex]\( \boxed{81} \)[/tex].
