Answer :
To solve the problem of finding [tex]\( f(1) \)[/tex] given the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and the information [tex]\( f(3) = 9 \)[/tex], we can follow these steps:
1. Understand the Recursive Function:
The function indicates that each term in the sequence is [tex]\(\frac{1}{3}\)[/tex] of the previous term. This means:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
2. Express [tex]\( f(3) \)[/tex] in terms of [tex]\( f(1) \)[/tex]:
Start with the expression for [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute the expression for [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} \left(\frac{1}{3} f(1)\right)
\][/tex]
Simplify the expression:
[tex]\[
f(3) = \left(\frac{1}{3}\right)^2 f(1) = \frac{1}{9} f(1)
\][/tex]
3. Use the given information [tex]\( f(3) = 9 \)[/tex]:
Substitute the known value of [tex]\( f(3) \)[/tex]:
[tex]\[
9 = \frac{1}{9} f(1)
\][/tex]
4. Solve for [tex]\( f(1) \)[/tex]:
Multiply both sides of the equation by 9 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 9 \times 9 = 81
\][/tex]
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
1. Understand the Recursive Function:
The function indicates that each term in the sequence is [tex]\(\frac{1}{3}\)[/tex] of the previous term. This means:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
2. Express [tex]\( f(3) \)[/tex] in terms of [tex]\( f(1) \)[/tex]:
Start with the expression for [tex]\( f(3) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute the expression for [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} \left(\frac{1}{3} f(1)\right)
\][/tex]
Simplify the expression:
[tex]\[
f(3) = \left(\frac{1}{3}\right)^2 f(1) = \frac{1}{9} f(1)
\][/tex]
3. Use the given information [tex]\( f(3) = 9 \)[/tex]:
Substitute the known value of [tex]\( f(3) \)[/tex]:
[tex]\[
9 = \frac{1}{9} f(1)
\][/tex]
4. Solve for [tex]\( f(1) \)[/tex]:
Multiply both sides of the equation by 9 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 9 \times 9 = 81
\][/tex]
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
