To solve the problem of predicting the next three numbers in the given pattern, we need to closely examine the relationship between the terms. The sequence we have is [tex]\(5\)[/tex], [tex]\(-10\)[/tex], [tex]\(20\)[/tex], [tex]\(-40\)[/tex].
First, observe the pattern:
1. The sign of each number alternates: positive, negative, positive, negative.
2. The magnitude of each successive number doubles compared to the previous number.
Starting with the last number in the provided sequence [tex]\(-40\)[/tex]:
1. To find the next number, we double the magnitude of [tex]\(-40\)[/tex] and change the sign to positive:
[tex]\[
-40 \times (-2) = 80
\][/tex]
So, the number following [tex]\(-40\)[/tex] is [tex]\(80\)[/tex].
2. Next, double the magnitude of [tex]\(80\)[/tex] and change the sign to negative:
[tex]\[
80 \times (-2) = -160
\][/tex]
So, the next number after [tex]\(80\)[/tex] is [tex]\(-160\)[/tex].
3. Finally, double the magnitude of [tex]\(-160\)[/tex] and change the sign to positive:
[tex]\[
-160 \times (-2) = 320
\][/tex]
So, the next number after [tex]\(-160\)[/tex] is [tex]\(320\)[/tex].
Thus, the next three numbers in the pattern [tex]\(5, -10, 20, -40\)[/tex] are:
[tex]\[ \boxed{80, -160, 320} \][/tex]
