Given the sequence defined by the recursive function [tex]\( f(n+1) = -10 f(n) \)[/tex] where [tex]\( f(1) = 1 \)[/tex], we aim to determine the value of [tex]\( f(3) \)[/tex].
Let's analyze the sequence step by step:
1. Initial Value:
The initial value given is [tex]\( f(1) = 1 \)[/tex].
2. First Recursion:
To find [tex]\( f(2) \)[/tex], we use the recursive definition:
[tex]\[
f(2) = -10 f(1)
\][/tex]
Substituting [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = -10 \times 1 = -10
\][/tex]
3. Second Recursion:
To find [tex]\( f(3) \)[/tex], we again use the recursive definition:
[tex]\[
f(3) = -10 f(2)
\][/tex]
Substituting [tex]\( f(2) \)[/tex]:
[tex]\[
f(3) = -10 \times (-10) = 100
\][/tex]
Through this process, we established that:
[tex]\[
f(1) = 1, \quad f(2) = -10, \quad f(3) = 100
\][/tex]
Thus, the value of [tex]\( f(3) \)[/tex] is:
[tex]\[
\boxed{100}
\][/tex]
