Answer :
To solve the problem, we need to find the value of [tex]\( f(1) \)[/tex] in 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. Understand the recursive definition: The sequence follows the rule [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term is one-third of the previous term.
2. Work backwards from [tex]\( f(3) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- To find [tex]\( f(2) \)[/tex], we use the inverse of the recursive function:
[tex]\[
f(n) = 3 \times f(n+1)
\][/tex]
So, [tex]\( f(2) = 3 \times f(3) = 3 \times 9 = 27 \)[/tex].
3. Find [tex]\( f(1) \)[/tex]:
- Now we find [tex]\( f(1) \)[/tex] using the same inverse process:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Let's break it down step-by-step:
1. Understand the recursive definition: The sequence follows the rule [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term is one-third of the previous term.
2. Work backwards from [tex]\( f(3) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- To find [tex]\( f(2) \)[/tex], we use the inverse of the recursive function:
[tex]\[
f(n) = 3 \times f(n+1)
\][/tex]
So, [tex]\( f(2) = 3 \times f(3) = 3 \times 9 = 27 \)[/tex].
3. Find [tex]\( f(1) \)[/tex]:
- Now we find [tex]\( f(1) \)[/tex] using the same inverse process:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
