Answer :
To solve the problem, we need to find [tex]\( f(1) \)[/tex] given the recursive sequence formula [tex]\( f(n+1) = \frac{1}{3}f(n) \)[/tex] and the fact that [tex]\( f(3) = 9 \)[/tex].
Let's work through the problem step-by-step:
1. Start with [tex]\( f(3) \)[/tex]:
We are given that [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
The recursive formula is [tex]\( f(n+1) = \frac{1}{3}f(n) \)[/tex]. To find the previous term [tex]\( f(2) \)[/tex], we need to reverse this operation.
So, multiply by 3:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Similarly, use the same process to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
Let's work through the problem step-by-step:
1. Start with [tex]\( f(3) \)[/tex]:
We are given that [tex]\( f(3) = 9 \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
The recursive formula is [tex]\( f(n+1) = \frac{1}{3}f(n) \)[/tex]. To find the previous term [tex]\( f(2) \)[/tex], we need to reverse this operation.
So, multiply by 3:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
Similarly, use the same process to find [tex]\( f(1) \)[/tex] from [tex]\( f(2) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is 81.
