Answer :
To solve this problem, let's follow the steps to find [tex]\( f(1) \)[/tex] given the recursive function and the value [tex]\( f(3) = 9 \)[/tex].
1. Understand the Recursive Function:
- The sequence is defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
- This means each term in the sequence is one-third of the previous term.
2. Use the Given Information:
- We know that [tex]\( f(3) = 9 \)[/tex].
3. Work Backwards to Find Previous Terms:
- Finding [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can find [tex]\( f(2) \)[/tex] by rearranging this equation:
[tex]\[
f(2) = 3 \times f(3)
\][/tex]
- Substitute [tex]\( f(3) = 9 \)[/tex] into the equation:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
- Finding [tex]\( f(1) \)[/tex]:
- Similarly, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], so we solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2)
\][/tex]
- Substitute [tex]\( f(2) = 27 \)[/tex] into the equation:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
4. Conclusion:
- The value of [tex]\( f(1) \)[/tex] is 81.
Therefore, the correct answer is [tex]\( \boxed{81} \)[/tex].
1. Understand the Recursive Function:
- The sequence is defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
- This means each term in the sequence is one-third of the previous term.
2. Use the Given Information:
- We know that [tex]\( f(3) = 9 \)[/tex].
3. Work Backwards to Find Previous Terms:
- Finding [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can find [tex]\( f(2) \)[/tex] by rearranging this equation:
[tex]\[
f(2) = 3 \times f(3)
\][/tex]
- Substitute [tex]\( f(3) = 9 \)[/tex] into the equation:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
- Finding [tex]\( f(1) \)[/tex]:
- Similarly, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], so we solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2)
\][/tex]
- Substitute [tex]\( f(2) = 27 \)[/tex] into the equation:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
4. Conclusion:
- The value of [tex]\( f(1) \)[/tex] is 81.
Therefore, the correct answer is [tex]\( \boxed{81} \)[/tex].
