Answer :
To solve for [tex]\( f(1) \)[/tex] in the given sequence, we need to work backwards from the information provided.
The sequence is defined by the recursive formula:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we know:
[tex]\[ f(3) = 9 \][/tex]
Let's find [tex]\( f(2) \)[/tex]:
From the recursive formula, we have:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Since [tex]\( f(3) = 9 \)[/tex], we can substitute this into the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]
Next, we find [tex]\( f(1) \)[/tex]:
Using the recursive formula again, we know:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex] into the equation:
[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 [tex]\( \boxed{81} \)[/tex].
The sequence is defined by the recursive formula:
[tex]\[ f(n+1) = \frac{1}{3} f(n) \][/tex]
And we know:
[tex]\[ f(3) = 9 \][/tex]
Let's find [tex]\( f(2) \)[/tex]:
From the recursive formula, we have:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Since [tex]\( f(3) = 9 \)[/tex], we can substitute this into the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]
Next, we find [tex]\( f(1) \)[/tex]:
Using the recursive formula again, we know:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Substitute [tex]\( f(2) = 27 \)[/tex] into the equation:
[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 [tex]\( \boxed{81} \)[/tex].
