High School

A sequence is defined by the recursive function [tex]f(n + 1) = \frac{1}{2} f(n)[/tex]. If [tex]f(3) = 9[/tex], what is [tex]f(1)[/tex]?

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

Answer :

The value of f(1) for the recursive function is (d) 81

The recursive function is given as:

[tex]f(n + 1) = \frac 13f(n)[/tex]

Multiply both sides of the equation by 3

[tex]3f(n + 1) = f(n)[/tex]

Rewrite as:

[tex]f(n) = 3f(n + 1)[/tex]

Set n = 2;

[tex]f(2) = 3f(2 + 1)[/tex]

[tex]f(2) = 3f(3)[/tex]

Set n = 1 in [tex]f(n) = 3f(n + 1)[/tex]

[tex]f(1) = 3f(1 + 1)[/tex]

[tex]f(1) = 3f(2)[/tex]

Substitute [tex]f(2) = 3f(3)[/tex]

[tex]f(1) = 3 \times 3f(3)[/tex]

[tex]f(1) = 9f(3)[/tex]

Substitute 9 for f(3)

[tex]f(1) = 9 \times 9[/tex]

[tex]f(1) = 81[/tex]

Hence, the value of f(1) for the recursive function is (d) 81

Read more about recursive functions at:

https://brainly.com/question/1275192

Answer:

81

Step-by-step explanation: