Answer :
To find [tex]\( f(1) \)[/tex] given the sequence defined by [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex] and knowing that [tex]\( f(3) = 9 \)[/tex], let's work backwards through the problem using the recursive formula.
1. Determine [tex]\( f(2) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- According to the recursive formula, [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
- To find [tex]\( f(2) \)[/tex], we'll use the inverse operation:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
2. Determine [tex]\( f(1) \)[/tex]:
- Now that we have [tex]\( f(2) = 27 \)[/tex], we can find [tex]\( f(1) \)[/tex] by again using the inverse of the recursive step:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
1. Determine [tex]\( f(2) \)[/tex]:
- We know [tex]\( f(3) = 9 \)[/tex].
- According to the recursive formula, [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex].
- To find [tex]\( f(2) \)[/tex], we'll use the inverse operation:
[tex]\[
f(2) = f(3) \times 3 = 9 \times 3 = 27
\][/tex]
2. Determine [tex]\( f(1) \)[/tex]:
- Now that we have [tex]\( f(2) = 27 \)[/tex], we can find [tex]\( f(1) \)[/tex] by again using the inverse of the recursive step:
[tex]\[
f(1) = f(2) \times 3 = 27 \times 3 = 81
\][/tex]
Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].
