Answer :
To solve the problem, we need to determine the initial term of the sequence, [tex]\( f(1) \)[/tex], using the recursive function and the given value [tex]\( f(3) = 9 \)[/tex].
1. Understanding the sequence:
The sequence is defined by the recursive formula:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
This means each term is one-third of the previous term.
2. Working backwards from [tex]\( f(3) = 9 \)[/tex]:
We know:
[tex]\[
f(3) = 9
\][/tex]
Using the recursive formula, we can express [tex]\( f(3) \)[/tex] in terms of [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substituting the given value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Determining [tex]\( f(1) \)[/tex]:
Similarly, using the recursive formula for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the value we found for [tex]\( f(2) \)[/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]
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
1. Understanding the sequence:
The sequence is defined by the recursive formula:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
This means each term is one-third of the previous term.
2. Working backwards from [tex]\( f(3) = 9 \)[/tex]:
We know:
[tex]\[
f(3) = 9
\][/tex]
Using the recursive formula, we can express [tex]\( f(3) \)[/tex] in terms of [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substituting the given value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Determining [tex]\( f(1) \)[/tex]:
Similarly, using the recursive formula for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the value we found for [tex]\( f(2) \)[/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]
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
