Answer :
To solve for [tex]\( f(1) \)[/tex] given the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and the value [tex]\( f(3) = 9 \)[/tex], follow these steps:
1. Understand the Recursive Relationship:
The sequence is defined by the relationship [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term is one-third of the previous term.
2. Work Backward from the Given Value:
We know that [tex]\( f(3) = 9 \)[/tex]. To find previous terms, we will work backwards using the given relationship. Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can find [tex]\( f(n) \)[/tex] by multiplying the next term by 3. Thus, [tex]\( f(n) = 3 \times f(n+1) \)[/tex].
3. Find [tex]\( f(2) \)[/tex]:
Using the relationship backward from [tex]\( f(3) \)[/tex], calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
4. Find [tex]\( f(1) \)[/tex]:
Now, use the relationship backward again from [tex]\( f(2) \)[/tex] to calculate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
1. Understand the Recursive Relationship:
The sequence is defined by the relationship [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex]. This means each term is one-third of the previous term.
2. Work Backward from the Given Value:
We know that [tex]\( f(3) = 9 \)[/tex]. To find previous terms, we will work backwards using the given relationship. Since [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can find [tex]\( f(n) \)[/tex] by multiplying the next term by 3. Thus, [tex]\( f(n) = 3 \times f(n+1) \)[/tex].
3. Find [tex]\( f(2) \)[/tex]:
Using the relationship backward from [tex]\( f(3) \)[/tex], calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 3 \times f(3) = 3 \times 9 = 27
\][/tex]
4. Find [tex]\( f(1) \)[/tex]:
Now, use the relationship backward again from [tex]\( f(2) \)[/tex] to calculate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 3 \times f(2) = 3 \times 27 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
