Answer :
To solve the problem, let's go through the sequence using the given information:
1. We are given the recursive function for the sequence: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
2. We also know that [tex]\( f(3) = 9 \)[/tex] and need to find [tex]\( f(1) \)[/tex].
Let's work backwards from what we know:
Step 1: From the definition [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Since [tex]\( f(3) = 9 \)[/tex], we substitute this into the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]
Step 2: Now, using the same recursive relation, we can express [tex]\( f(1) \)[/tex] in terms of [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Substitute the value of [tex]\( f(2) = 27 \)[/tex] into the equation:
[tex]\[ 27 = \frac{1}{3} f(1) \][/tex]
To solve for [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[ f(1) = 27 \times 3 = 81 \][/tex]
Therefore, [tex]\( f(1) = 81 \)[/tex].
1. We are given the recursive function for the sequence: [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
2. We also know that [tex]\( f(3) = 9 \)[/tex] and need to find [tex]\( f(1) \)[/tex].
Let's work backwards from what we know:
Step 1: From the definition [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], we can express [tex]\( f(2) \)[/tex] in terms of [tex]\( f(3) \)[/tex]:
[tex]\[ f(3) = \frac{1}{3} f(2) \][/tex]
Since [tex]\( f(3) = 9 \)[/tex], we substitute this into the equation:
[tex]\[ 9 = \frac{1}{3} f(2) \][/tex]
To solve for [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[ f(2) = 9 \times 3 = 27 \][/tex]
Step 2: Now, using the same recursive relation, we can express [tex]\( f(1) \)[/tex] in terms of [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = \frac{1}{3} f(1) \][/tex]
Substitute the value of [tex]\( f(2) = 27 \)[/tex] into the equation:
[tex]\[ 27 = \frac{1}{3} f(1) \][/tex]
To solve for [tex]\( f(1) \)[/tex], multiply both sides by 3:
[tex]\[ f(1) = 27 \times 3 = 81 \][/tex]
Therefore, [tex]\( f(1) = 81 \)[/tex].
