Answer :
To solve this problem, we need to find [tex]\( f(1) \)[/tex] given that the sequence is defined recursively by [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and [tex]\( f(3) = 9 \)[/tex].
Let's work our way backwards from [tex]\( f(3) \)[/tex] to [tex]\( f(1) \)[/tex].
1. Given Information:
- We know that [tex]\( f(3) = 9 \)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
- We use the recursive formula: [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- Solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Again, using the recursive formula: [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- Solving for [tex]\( f(1) \)[/tex], we multiply both sides by 3:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Thus, [tex]\( f(1) = 81 \)[/tex]. The correct answer is 81.
Let's work our way backwards from [tex]\( f(3) \)[/tex] to [tex]\( f(1) \)[/tex].
1. Given Information:
- We know that [tex]\( f(3) = 9 \)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
- We use the recursive formula: [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- Solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Again, using the recursive formula: [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- Solving for [tex]\( f(1) \)[/tex], we multiply both sides by 3:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Thus, [tex]\( f(1) = 81 \)[/tex]. The correct answer is 81.
