Answer :
To solve this problem, we need to find the value of [tex]\( f(1) \)[/tex] given the recursive sequence defined by [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and the condition [tex]\( f(3) = 9 \)[/tex].
Let's go step-by-step:
1. Understand the Recursive Formula:
- The 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. Working Backwards:
- We know [tex]\( f(3) = 9 \)[/tex]. We want to find [tex]\( f(1) \)[/tex].
3. Find [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can rearrange this formula 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]:
- Now, use the same process to find [tex]\( f(1) \)[/tex].
- From the recursive formula [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], solve for [tex]\( f(1) \)[/tex] by multiplying both sides by 3:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
Let's go step-by-step:
1. Understand the Recursive Formula:
- The 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. Working Backwards:
- We know [tex]\( f(3) = 9 \)[/tex]. We want to find [tex]\( f(1) \)[/tex].
3. Find [tex]\( f(2) \)[/tex]:
- Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can rearrange this formula 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]:
- Now, use the same process to find [tex]\( f(1) \)[/tex].
- From the recursive formula [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex], solve for [tex]\( f(1) \)[/tex] by multiplying both sides by 3:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
