Answer :
To solve this problem, we need to find the value of [tex]\( f(1) \)[/tex] based on the given recursive relationship and information.
1. Understand the Recursive Function:
The recursive function is defined as [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term in the sequence is one-third of the previous term.
2. Given Information:
We are provided with [tex]\( f(3) = 9 \)[/tex].
3. Work Backwards to Find [tex]\( f(1) \)[/tex]:
- Determine [tex]\( f(2) \)[/tex]:
Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can express this as:
[tex]\[
f(2) = 3 \times f(3)
\][/tex]
Substitute the given [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
- Determine [tex]\( f(1) \)[/tex]:
Similarly, using the recursive definition for the previous term:
[tex]\[
f(1) = 3 \times f(2)
\][/tex]
Substitute the known value [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
1. Understand the Recursive Function:
The recursive function is defined as [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term in the sequence is one-third of the previous term.
2. Given Information:
We are provided with [tex]\( f(3) = 9 \)[/tex].
3. Work Backwards to Find [tex]\( f(1) \)[/tex]:
- Determine [tex]\( f(2) \)[/tex]:
Since [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex], we can express this as:
[tex]\[
f(2) = 3 \times f(3)
\][/tex]
Substitute the given [tex]\( f(3) = 9 \)[/tex]:
[tex]\[
f(2) = 3 \times 9 = 27
\][/tex]
- Determine [tex]\( f(1) \)[/tex]:
Similarly, using the recursive definition for the previous term:
[tex]\[
f(1) = 3 \times f(2)
\][/tex]
Substitute the known value [tex]\( f(2) = 27 \)[/tex]:
[tex]\[
f(1) = 3 \times 27 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
