Answer :
Sure! Let's solve the problem step-by-step.
We are given a recursive function for a sequence: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
We also know that [tex]\( f(3) = 9 \)[/tex].
We want to find [tex]\( f(1) \)[/tex].
Step 1: Express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(3) \)[/tex]
From the recursive relationship, we have:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
To find [tex]\( f(2) \)[/tex], we rearrange this to:
[tex]\[ f(2) = 3 \times f(3) \][/tex]
Given [tex]\( f(3) = 9 \)[/tex], substitute this value in:
[tex]\[ f(2) = 3 \times 9 = 27 \][/tex]
Step 2: Express [tex]\( f(1) \)[/tex] in terms of [tex]\( f(2) \)[/tex]
Using the recursive relationship again:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
To find [tex]\( f(1) \)[/tex], rearrange this to:
[tex]\[ f(1) = 3 \times f(2) \][/tex]
Now substitute [tex]\( f(2) = 27 \)[/tex] in:
[tex]\[ f(1) = 3 \times 27 = 81 \][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
We are given a recursive function for a sequence: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
We also know that [tex]\( f(3) = 9 \)[/tex].
We want to find [tex]\( f(1) \)[/tex].
Step 1: Express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(3) \)[/tex]
From the recursive relationship, we have:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
To find [tex]\( f(2) \)[/tex], we rearrange this to:
[tex]\[ f(2) = 3 \times f(3) \][/tex]
Given [tex]\( f(3) = 9 \)[/tex], substitute this value in:
[tex]\[ f(2) = 3 \times 9 = 27 \][/tex]
Step 2: Express [tex]\( f(1) \)[/tex] in terms of [tex]\( f(2) \)[/tex]
Using the recursive relationship again:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
To find [tex]\( f(1) \)[/tex], rearrange this to:
[tex]\[ f(1) = 3 \times f(2) \][/tex]
Now substitute [tex]\( f(2) = 27 \)[/tex] in:
[tex]\[ f(1) = 3 \times 27 = 81 \][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
