Answer :
To solve this problem, we are given that [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and [tex]\( f(3) = 9 \)[/tex]. We need to find [tex]\( f(1) \)[/tex].
The recursive formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] indicates how we can compute the next term based on the previous term. Since it's a recursive function headed forward through the indices, we can also work backwards by reversing the operation to find previous terms from subsequent terms.
1. Step to find [tex]\( f(2) \)[/tex]:
- From the equation, moving backward, multiply the term by 3. So, we have:
- [tex]\( f(2) = 3 \times f(3) \)[/tex]
- Since [tex]\( f(3) = 9 \)[/tex], then:
- [tex]\( f(2) = 3 \times 9 = 27 \)[/tex]
2. Step to find [tex]\( f(1) \)[/tex]:
- Similarly, multiply the previous [tex]\( f(2) \)[/tex] term by 3:
- [tex]\( f(1) = 3 \times f(2) \)[/tex]
- Since [tex]\( f(2) = 27 \)[/tex], then:
- [tex]\( f(1) = 3 \times 27 = 81 \)[/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
The recursive formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] indicates how we can compute the next term based on the previous term. Since it's a recursive function headed forward through the indices, we can also work backwards by reversing the operation to find previous terms from subsequent terms.
1. Step to find [tex]\( f(2) \)[/tex]:
- From the equation, moving backward, multiply the term by 3. So, we have:
- [tex]\( f(2) = 3 \times f(3) \)[/tex]
- Since [tex]\( f(3) = 9 \)[/tex], then:
- [tex]\( f(2) = 3 \times 9 = 27 \)[/tex]
2. Step to find [tex]\( f(1) \)[/tex]:
- Similarly, multiply the previous [tex]\( f(2) \)[/tex] term by 3:
- [tex]\( f(1) = 3 \times f(2) \)[/tex]
- Since [tex]\( f(2) = 27 \)[/tex], then:
- [tex]\( f(1) = 3 \times 27 = 81 \)[/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
