Feedback Formula

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

The Formula

xn+1=f(xn)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

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

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

Worked Examples

Example 1

medium
Iterate the map xn+1=0.5xn+3x_{n+1} = 0.5x_n + 3 starting from x0=10x_0 = 10. Compute x1,x2,x3x_1, x_2, x_3 and predict the long-run value.

Answer

x1=8,x2=7,x3=6.5x_1=8, x_2=7, x_3=6.5; long-run value x∗=6x^*=6

First step

1
x1=0.5(10)+3=8x_1 = 0.5(10)+3 = 8; x2=0.5(8)+3=7x_2 = 0.5(8)+3 = 7; x3=0.5(7)+3=6.5x_3 = 0.5(7)+3 = 6.5.

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard
Analyze the logistic map xn+1=3.5xn(1−xn)x_{n+1} = 3.5 x_n(1-x_n) by iterating from x0=0.5x_0=0.5 for five steps and commenting on the behavior.

Example 3

medium
Find the fixed point of xn+1=0.6xn+4x_{n+1} = 0.6 x_n + 4.

Common Mistakes

  • Swapping positive and negative feedback - positive amplifies/destabilizes, negative corrects/stabilizes.
  • Treating a feedback loop like a one-shot function - the rule is applied repeatedly, each output re-entering as input.
  • Assuming positive feedback always grows without limit - real loops often hit saturation or get checked by negative feedback.

Why This Formula Matters

Feedback is the engine behind runaway growth, thermostats, population dynamics, and chaos — anywhere today's value sets tomorrow's. Seeing the loop lets a student iterate a recurrence and predict whether a system blows up, settles, or oscillates instead of treating each step in isolation. Recognizing it by "Does the system's output get fed back in as the input for the next step?" — rather than by familiar numbers — is what lets a student tell it apart from composition chains and stability of an equilibrium and recurrence / iteration in a mixed problem set.

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?

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

Why is the Feedback formula important in Math?

Feedback is the engine behind runaway growth, thermostats, population dynamics, and chaos — anywhere today's value sets tomorrow's. Seeing the loop lets a student iterate a recurrence and predict whether a system blows up, settles, or oscillates instead of treating each step in isolation. Recognizing it by "Does the system's output get fed back in as the input for the next step?" — rather than by familiar numbers — is what lets a student tell it apart from composition chains and stability of an equilibrium and recurrence / iteration in a mixed problem set.

What do students get wrong about Feedback?

The procedure for feedback is the easy part; the trap is swapping positive and negative feedback. Asking "Does the system's output get fed back in as the input for the next step?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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