Feedback Formula

The Formula

x_{n+1} = f(x_n) (output feeds back as the next input)

When to use: Microphone feedback: sound → speaker → microphone → more sound → louder...

Quick Example

Compound interest: interest earns interest. More money → more interest → even more money.

Notation

x_{n+1} = f(x_n) denotes a recurrence where the output of step n becomes the input of step n+1.

What This Formula Means

Feedback occurs when the output of a system influences its future input — positive feedback amplifies changes; negative feedback stabilizes them.

Microphone feedback: sound → speaker → microphone → more sound → louder...

Formal View

x_{n+1} = f(x_n); positive feedback: |f'(x^*)| > 1 (amplifies perturbations); negative feedback: |f'(x^*)| < 1 (dampens perturbations near equilibrium x^*)

Worked Examples

Example 1

medium
Iterate the map x_{n+1} = 0.5x_n + 3 starting from x_0 = 10. Compute x_1, x_2, x_3 and predict the long-run value.

Solution

  1. 1
    x_1 = 0.5(10)+3 = 8; x_2 = 0.5(8)+3 = 7; x_3 = 0.5(7)+3 = 6.5.
  2. 2
    The sequence appears to converge. Fixed point: solve x^* = 0.5x^*+3 \Rightarrow 0.5x^*=3 \Rightarrow x^*=6.
  3. 3
    Since |f'(x)|=|0.5|<1 at the fixed point, x^*=6 is stable. The sequence converges to 6.

Answer

x_1=8, x_2=7, x_3=6.5; long-run value x^*=6
A feedback map repeatedly applies a function to its previous output. If the map has a stable fixed point, iteration converges there regardless of starting value (within the basin of attraction).

Example 2

hard
Analyze the logistic map x_{n+1} = 3.5 x_n(1-x_n) by iterating from x_0=0.5 for five steps and commenting on the behavior.

Common Mistakes

  • Thinking all feedback is positive (amplifying) — negative feedback dampens and stabilizes; positive feedback amplifies
  • Confusing feedback with simple cause-and-effect — feedback is specifically when the output loops back to influence the input
  • Assuming feedback always leads to explosion — negative feedback creates stable equilibria, not runaway behavior

Why This Formula Matters

Feedback explains explosive growth and self-regulating systems.

Frequently Asked Questions

What is the Feedback formula?

Feedback occurs when the output of a system influences its future input — positive feedback amplifies changes; negative feedback stabilizes them.

How do you use the Feedback formula?

Microphone feedback: sound → speaker → microphone → more sound → louder...

What do the symbols mean in the Feedback formula?

x_{n+1} = f(x_n) denotes a recurrence where the output of step n becomes the input of step n+1.

Why is the Feedback formula important in Math?

Feedback explains explosive growth and self-regulating systems.

What do students get wrong about Feedback?

"Positive feedback" does not mean "good feedback" — it means the feedback reinforces change, which can be destabilizing.

What should I learn before the Feedback formula?

Before studying the Feedback formula, you should understand: exponential function.

← Back to Feedback See All Examples Practice Problems