Answer :
To solve the problem, we need to find the value of [tex]\( f(1) \)[/tex] for the given recursive sequence. The sequence is defined as [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. We also know that [tex]\( f(3) = 9 \)[/tex].
Let's work backwards from the information provided:
1. Find [tex]\( f(2) \)[/tex]:
We have the equation:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Next, use the equation:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To solve for [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
Let's work backwards from the information provided:
1. Find [tex]\( f(2) \)[/tex]:
We have the equation:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substitute [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
2. Find [tex]\( f(1) \)[/tex]:
Next, use the equation:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
To solve for [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
