Answer :
To solve the problem, we need to find the initial term, [tex]\( f(1) \)[/tex], of the sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], given that [tex]\( f(3) = 9 \)[/tex].
Let's work through the recursive steps:
1. Identify the sequence relationship:
- The relationship between consecutive terms is given by the formula [tex]\( f(n+1) = \frac{1}{3} \times f(n) \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
- We have the relationship [tex]\( f(3) = \frac{1}{3} \times f(2) \)[/tex].
- Plugging in [tex]\( f(3) = 9 \)[/tex], we get:
[tex]\[
9 = \frac{1}{3} \times f(2)
\][/tex]
- Solving for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
- Use the formula again: [tex]\( f(2) = \frac{1}{3} \times f(1) \)[/tex].
- We have just calculated [tex]\( f(2) = 27 \)[/tex], therefore:
[tex]\[
27 = \frac{1}{3} \times f(1)
\][/tex]
- To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
So, the initial term of the sequence, [tex]\( f(1) \)[/tex], is [tex]\(\boxed{81}\)[/tex].
Let's work through the recursive steps:
1. Identify the sequence relationship:
- The relationship between consecutive terms is given by the formula [tex]\( f(n+1) = \frac{1}{3} \times f(n) \)[/tex].
2. Find [tex]\( f(2) \)[/tex]:
- We have the relationship [tex]\( f(3) = \frac{1}{3} \times f(2) \)[/tex].
- Plugging in [tex]\( f(3) = 9 \)[/tex], we get:
[tex]\[
9 = \frac{1}{3} \times f(2)
\][/tex]
- Solving for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Find [tex]\( f(1) \)[/tex]:
- Use the formula again: [tex]\( f(2) = \frac{1}{3} \times f(1) \)[/tex].
- We have just calculated [tex]\( f(2) = 27 \)[/tex], therefore:
[tex]\[
27 = \frac{1}{3} \times f(1)
\][/tex]
- To find [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
So, the initial term of the sequence, [tex]\( f(1) \)[/tex], is [tex]\(\boxed{81}\)[/tex].
