Answer :
To solve the given problem, we need to understand the recursive function for the sequence. The sequence is defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. We are given that [tex]\( f(3) = 9 \)[/tex] and we need to find [tex]\( f(1) \)[/tex].
Let's work through this step by step:
1. Start from the known value:
We know that [tex]\( f(3) = 9 \)[/tex].
2. Calculate [tex]\( f(2) \)[/tex]:
Since the recursive function is [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can express [tex]\( f(n) \)[/tex] in terms of [tex]\( f(n+1) \)[/tex] as:
[tex]\[
f(n) = 3 \cdot f(n+1)
\][/tex]
So using [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Calculate [tex]\( f(1) \)[/tex]:
Similarly, using the same transformation to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
Let's work through this step by step:
1. Start from the known value:
We know that [tex]\( f(3) = 9 \)[/tex].
2. Calculate [tex]\( f(2) \)[/tex]:
Since the recursive function is [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can express [tex]\( f(n) \)[/tex] in terms of [tex]\( f(n+1) \)[/tex] as:
[tex]\[
f(n) = 3 \cdot f(n+1)
\][/tex]
So using [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Calculate [tex]\( f(1) \)[/tex]:
Similarly, using the same transformation to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Thus, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
