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 break it down step-by-step:
1. Understanding the Recursive Formula:
- The sequence is given by [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
- This means each term is one-third of the previous term.
2. Finding [tex]\( f(2) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- To find [tex]\( f(2) \)[/tex], we use the recursive formula:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
So, we can solve for [tex]\( f(2) \)[/tex] by rearranging the equation:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Now, use the same process to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex], we get:
[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. Understanding the Recursive Formula:
- The sequence is given by [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
- This means each term is one-third of the previous term.
2. Finding [tex]\( f(2) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- To find [tex]\( f(2) \)[/tex], we use the recursive formula:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
So, we can solve for [tex]\( f(2) \)[/tex] by rearranging the equation:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Now, use the same process to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex], we get:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
