Answer :
To solve the problem, we need to find [tex]\( f(1) \)[/tex] given the recursive function and the value [tex]\( f(3) = 9 \)[/tex].
1. Understanding the Recurrence:
The sequence is defined by the recursive function:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
This means that each term in the sequence is one-third of the previous term.
2. Work Backwards from [tex]\( f(3) \)[/tex]:
Since we're told that [tex]\( f(3) = 9 \)[/tex], let's use the recursion to find earlier terms:
- Finding [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2) \Rightarrow 9 = \frac{1}{3} f(2)
\][/tex]
Solve for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
- Finding [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1) \Rightarrow 27 = \frac{1}{3} f(1)
\][/tex]
Solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
3. Conclusion:
Thus, the value of [tex]\( f(1) \)[/tex] is 81.
1. Understanding the Recurrence:
The sequence is defined by the recursive function:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
This means that each term in the sequence is one-third of the previous term.
2. Work Backwards from [tex]\( f(3) \)[/tex]:
Since we're told that [tex]\( f(3) = 9 \)[/tex], let's use the recursion to find earlier terms:
- Finding [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = \frac{1}{3} f(2) \Rightarrow 9 = \frac{1}{3} f(2)
\][/tex]
Solve for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
- Finding [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1) \Rightarrow 27 = \frac{1}{3} f(1)
\][/tex]
Solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
3. Conclusion:
Thus, the value of [tex]\( f(1) \)[/tex] is 81.
