Answer :
To solve the problem, we are given a sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and we know that [tex]\( f(3) = 9 \)[/tex]. Our goal is to find [tex]\( f(1) \)[/tex].
Here's how we can approach this step-by-step:
1. Understand the recursive formula:
- The given formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] tells us that each term in the sequence is one-third of the previous term.
2. Work backwards from [tex]\( f(3) \)[/tex]:
- Start from [tex]\( f(3) = 9 \)[/tex] and use the recursive relationship to find earlier terms.
3. Find [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can rearrange this to find [tex]\( f(2) \)[/tex].
- Multiply both sides by 3 to solve for [tex]\( f(2) \)[/tex]: [tex]\( f(2) = 3 \times f(3) = 3 \times 9 = 27 \)[/tex].
4. Find [tex]\( f(1) \)[/tex]:
- Similarly, use the relationship [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- Rearrange and solve for [tex]\( f(1) \)[/tex]: [tex]\( f(1) = 3 \times f(2) = 3 \times 27 = 81 \)[/tex].
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
Here's how we can approach this step-by-step:
1. Understand the recursive formula:
- The given formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] tells us that each term in the sequence is one-third of the previous term.
2. Work backwards from [tex]\( f(3) \)[/tex]:
- Start from [tex]\( f(3) = 9 \)[/tex] and use the recursive relationship to find earlier terms.
3. Find [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can rearrange this to find [tex]\( f(2) \)[/tex].
- Multiply both sides by 3 to solve for [tex]\( f(2) \)[/tex]: [tex]\( f(2) = 3 \times f(3) = 3 \times 9 = 27 \)[/tex].
4. Find [tex]\( f(1) \)[/tex]:
- Similarly, use the relationship [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- Rearrange and solve for [tex]\( f(1) \)[/tex]: [tex]\( f(1) = 3 \times f(2) = 3 \times 27 = 81 \)[/tex].
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
