Answer :
To solve the problem, we need to determine the value of [tex]\( f(1) \)[/tex] given the recursive relationship and a specific term in the sequence. Let's break it down step-by-step.
1. Understand the Recursive Relationship:
The sequence is defined recursively as [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term is one-third of the previous term.
2. Given Information:
We know that [tex]\( f(3) = 9 \)[/tex].
3. Find [tex]\( f(2) \)[/tex]:
According to the recursive relationship:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute the known value [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 27
\][/tex]
4. Find [tex]\( f(1) \)[/tex]:
Use the recursive relationship again:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the known value [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
1. Understand the Recursive Relationship:
The sequence is defined recursively as [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term is one-third of the previous term.
2. Given Information:
We know that [tex]\( f(3) = 9 \)[/tex].
3. Find [tex]\( f(2) \)[/tex]:
According to the recursive relationship:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute the known value [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 27
\][/tex]
4. Find [tex]\( f(1) \)[/tex]:
Use the recursive relationship again:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute the known value [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
