Answer :
To find [tex]\( f(1) \)[/tex] in the sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], and given that [tex]\( f(3) = 9 \)[/tex], we can work backwards using the recursive relationship.
1. Determine [tex]\( f(2) \)[/tex]:
From the recursive function, we know that:
[tex]\[
f(3) = \frac{1}{3}f(2)
\][/tex]
Since [tex]\( f(3) = 9 \)[/tex], we substitute and solve for [tex]\( f(2) \)[/tex]:
[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]
2. Determine [tex]\( f(1) \)[/tex]:
Use the recursive relationship again for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3}f(1)
\][/tex]
With [tex]\( f(2) = 27 \)[/tex], substitute to find [tex]\( f(1) \)[/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].
1. Determine [tex]\( f(2) \)[/tex]:
From the recursive function, we know that:
[tex]\[
f(3) = \frac{1}{3}f(2)
\][/tex]
Since [tex]\( f(3) = 9 \)[/tex], we substitute and solve for [tex]\( f(2) \)[/tex]:
[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]
2. Determine [tex]\( f(1) \)[/tex]:
Use the recursive relationship again for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3}f(1)
\][/tex]
With [tex]\( f(2) = 27 \)[/tex], substitute to find [tex]\( f(1) \)[/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].
