Answer :
To solve this problem, we need to find the value of [tex]\( f(1) \)[/tex] given that the sequence is defined by a recursive relation and [tex]\( f(3) = 9 \)[/tex].
The recursive function given is [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that each term in the sequence is one-third of the previous term.
Let's work backward from [tex]\( f(3) \)[/tex] to find [tex]\( f(1) \)[/tex].
1. Finding [tex]\( f(2) \)[/tex]:
We know that [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex]. To express this in terms of [tex]\( f(2) \)[/tex], rearrange it as:
[tex]\[
f(2) = 3 \cdot f(3)
\][/tex]
Since [tex]\( f(3) = 9 \)[/tex], substitute:
[tex]\[
f(2) = 3 \cdot 9 = 27
\][/tex]
2. Finding [tex]\( f(1) \)[/tex]:
Now, use the same logic to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Rearrange it to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \cdot f(2)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
f(1) = 3 \cdot 27 = 81
\][/tex]
The value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
The recursive function given is [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means that each term in the sequence is one-third of the previous term.
Let's work backward from [tex]\( f(3) \)[/tex] to find [tex]\( f(1) \)[/tex].
1. Finding [tex]\( f(2) \)[/tex]:
We know that [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex]. To express this in terms of [tex]\( f(2) \)[/tex], rearrange it as:
[tex]\[
f(2) = 3 \cdot f(3)
\][/tex]
Since [tex]\( f(3) = 9 \)[/tex], substitute:
[tex]\[
f(2) = 3 \cdot 9 = 27
\][/tex]
2. Finding [tex]\( f(1) \)[/tex]:
Now, use the same logic to find [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Rearrange it to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \cdot f(2)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
f(1) = 3 \cdot 27 = 81
\][/tex]
The value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
