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A sequence is defined by the recursive function [tex]f(n+1)=\frac{1}{3} f(n)[/tex]. If [tex]f(3)=9[/tex], what is [tex]f(1)[/tex]?

A. 1
B. 3
C. 27
D. 81

Answer :

To find [tex]\( f(1) \)[/tex] in the sequence defined by the recursive function [tex]\( f(n+1) = \frac{1}{3} f(n) \)[/tex], and given that [tex]\( f(3) = 9 \)[/tex], we can work backwards using the recursive relationship.

1. Determine [tex]\( f(2) \)[/tex]:

From the recursive function, we know that:
[tex]\[
f(3) = \frac{1}{3}f(2)
\][/tex]

Since [tex]\( f(3) = 9 \)[/tex], we substitute and solve for [tex]\( f(2) \)[/tex]:
[tex]\[
9 = \frac{1}{3}f(2)
\][/tex]

To find [tex]\( f(2) \)[/tex], multiply both sides by 3:
[tex]\[
f(2) = 9 \times 3 = 27
\][/tex]

2. Determine [tex]\( f(1) \)[/tex]:

Use the recursive relationship again for [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = \frac{1}{3}f(1)
\][/tex]

With [tex]\( f(2) = 27 \)[/tex], substitute to find [tex]\( f(1) \)[/tex]:
[tex]\[
27 = \frac{1}{3}f(1)
\][/tex]

Multiply both sides by 3 to solve for [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 27 \times 3 = 81
\][/tex]

Therefore, the value of [tex]\( f(1) \)[/tex] is [tex]\( 81 \)[/tex].

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