Answer :
To find [tex]\( f(1) \)[/tex] in the given sequence, follow these steps based on the recursive formula:
1. We begin with the given recursive function:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
2. We know that [tex]\( f(3) = 9 \)[/tex].
3. We first need to find [tex]\( f(2) \)[/tex]. From the recursive function, we have:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substituting the known value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
4. Next, we find [tex]\( f(1) \)[/tex]. Using the recursive formula again for [tex]\( n = 1 \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substituting the value we found for [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex], we multiply both sides by 3:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
1. We begin with the given recursive function:
[tex]\[
f(n+1) = \frac{1}{3} f(n)
\][/tex]
2. We know that [tex]\( f(3) = 9 \)[/tex].
3. We first need to find [tex]\( f(2) \)[/tex]. From the recursive function, we have:
[tex]\[
f(3) = \frac{1}{3} f(2)
\][/tex]
Substituting the known value:
[tex]\[
9 = \frac{1}{3} f(2)
\][/tex]
Solving for [tex]\( f(2) \)[/tex], we multiply both sides by 3:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
4. Next, we find [tex]\( f(1) \)[/tex]. Using the recursive formula again for [tex]\( n = 1 \)[/tex]:
[tex]\[
f(2) = \frac{1}{3} f(1)
\][/tex]
Substituting the value we found for [tex]\( f(2) \)[/tex]:
[tex]\[
27 = \frac{1}{3} f(1)
\][/tex]
Solving for [tex]\( f(1) \)[/tex], we multiply both sides by 3:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\(\boxed{81}\)[/tex].
