Answer :
To solve the given problem, we need to determine the value of [tex]\( f(1) \)[/tex] for the sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], given that [tex]\( f(3) = 9 \)[/tex].
Let's break it down step by step:
1. Identify the recursive relationship: The sequence is defined by [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that the value of the next term in the sequence is one-third of the current term.
2. Determine [tex]\( f(2) \)[/tex] using [tex]\( f(3) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- According to the recursive relationship, [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- To find [tex]\( f(2) \)[/tex], we rearrange the equation: [tex]\( f(2) = 3 \times f(3) \)[/tex].
- Substituting [tex]\( f(3) = 9 \)[/tex]: [tex]\( f(2) = 3 \times 9 = 27 \)[/tex].
3. Determine [tex]\( f(1) \)[/tex] using [tex]\( f(2) \)[/tex]:
- We now know [tex]\( f(2) = 27 \)[/tex].
- According to the recursive relationship, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- To find [tex]\( f(1) \)[/tex], we rearrange the equation: [tex]\( f(1) = 3 \times f(2) \)[/tex].
- Substituting [tex]\( f(2) = 27 \)[/tex]: [tex]\( f(1) = 3 \times 27 = 81 \)[/tex].
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\( \boxed{81} \)[/tex].
Let's break it down step by step:
1. Identify the recursive relationship: The sequence is defined by [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that the value of the next term in the sequence is one-third of the current term.
2. Determine [tex]\( f(2) \)[/tex] using [tex]\( f(3) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- According to the recursive relationship, [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- To find [tex]\( f(2) \)[/tex], we rearrange the equation: [tex]\( f(2) = 3 \times f(3) \)[/tex].
- Substituting [tex]\( f(3) = 9 \)[/tex]: [tex]\( f(2) = 3 \times 9 = 27 \)[/tex].
3. Determine [tex]\( f(1) \)[/tex] using [tex]\( f(2) \)[/tex]:
- We now know [tex]\( f(2) = 27 \)[/tex].
- According to the recursive relationship, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- To find [tex]\( f(1) \)[/tex], we rearrange the equation: [tex]\( f(1) = 3 \times f(2) \)[/tex].
- Substituting [tex]\( f(2) = 27 \)[/tex]: [tex]\( f(1) = 3 \times 27 = 81 \)[/tex].
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\( \boxed{81} \)[/tex].
