Answer :
To solve the problem, we need to find the value of [tex]\( f(1) \)[/tex] given the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and knowing that [tex]\( f(3) = 9 \)[/tex].
Here's a step-by-step solution:
1. Understanding the Recursive Relationship:
- The sequence is defined such that each term is one-third of the previous term: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
- From the relationship, we know [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- We are given that [tex]\( f(3) = 9 \)[/tex].
- To find [tex]\( f(2) \)[/tex], multiply [tex]\( f(3) \)[/tex] by 3 to reverse the operation:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Similarly, from the recursive formula, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- We have calculated [tex]\( f(2) = 27 \)[/tex].
- To find [tex]\( f(1) \)[/tex], multiply [tex]\( f(2) \)[/tex] by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is 81.
Here's a step-by-step solution:
1. Understanding the Recursive Relationship:
- The sequence is defined such that each term is one-third of the previous term: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
2. Finding [tex]\( f(2) \)[/tex]:
- From the relationship, we know [tex]\( f(3) = \frac{1}{3} f(2) \)[/tex].
- We are given that [tex]\( f(3) = 9 \)[/tex].
- To find [tex]\( f(2) \)[/tex], multiply [tex]\( f(3) \)[/tex] by 3 to reverse the operation:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]
3. Finding [tex]\( f(1) \)[/tex]:
- Similarly, from the recursive formula, [tex]\( f(2) = \frac{1}{3} f(1) \)[/tex].
- We have calculated [tex]\( f(2) = 27 \)[/tex].
- To find [tex]\( f(1) \)[/tex], multiply [tex]\( f(2) \)[/tex] by 3:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]
So, the value of [tex]\( f(1) \)[/tex] is 81.
