Answer :
To solve this problem, we need to determine the value of [tex]\( f(1) \)[/tex] given the recursive formula for the sequence and the fact that [tex]\( f(3) = 9 \)[/tex].
The recursive relationship given is:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
We need to use this relationship to find previous terms in the sequence, starting from [tex]\( f(3) \)[/tex] and working backward to find [tex]\( f(1) \)[/tex].
1. Find [tex]\( f(2) \)[/tex]:
- We know [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- Therefore, [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].
2. Find [tex]\( f(1) \)[/tex]:
- Next, use the formula to express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(1) \)[/tex]:
- [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- Therefore, [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].
Thus, the value of [tex]\( f(1) \)[/tex] is 81.
The recursive relationship given is:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
We need to use this relationship to find previous terms in the sequence, starting from [tex]\( f(3) \)[/tex] and working backward to find [tex]\( f(1) \)[/tex].
1. Find [tex]\( f(2) \)[/tex]:
- We know [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- Therefore, [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].
2. Find [tex]\( f(1) \)[/tex]:
- Next, use the formula to express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(1) \)[/tex]:
- [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- Therefore, [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].
Thus, the value of [tex]\( f(1) \)[/tex] is 81.
