Answer :
To solve this problem, we have a recursive sequence defined by the formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. We know that [tex]\( f(3) = 9 \)[/tex] and we need to find [tex]\( f(1) \)[/tex].
Let's break this down step by step:
1. Determine [tex]\( f(2) \)[/tex]:
- We are given that [tex]\( f(3) = 9 \)[/tex].
- According to the recursive formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can find [tex]\( f(2) \)[/tex] as follows:
[tex]\[
f(3) = \frac{1}{3} f(2) \Rightarrow 9 = \frac{1}{3} f(2)
\][/tex]
- To solve for [tex]\( f(2) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Determine [tex]\( f(1) \)[/tex]:
- Next, use the recursive relation again to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1) \Rightarrow 27 = \frac{1}{3} f(1)
\][/tex]
- Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
So, through these steps, we find that [tex]\( f(1) = 81 \)[/tex].
Let's break this down step by step:
1. Determine [tex]\( f(2) \)[/tex]:
- We are given that [tex]\( f(3) = 9 \)[/tex].
- According to the recursive formula [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can find [tex]\( f(2) \)[/tex] as follows:
[tex]\[
f(3) = \frac{1}{3} f(2) \Rightarrow 9 = \frac{1}{3} f(2)
\][/tex]
- To solve for [tex]\( f(2) \)[/tex], multiply both sides of the equation by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Determine [tex]\( f(1) \)[/tex]:
- Next, use the recursive relation again to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1) \Rightarrow 27 = \frac{1}{3} f(1)
\][/tex]
- Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
So, through these steps, we find that [tex]\( f(1) = 81 \)[/tex].
