Rukasha Maths

GCSE maths Beta

Iteration and numerical solutions · Higher notes

Higher · Algebra · AQA references A20, R16

Iteration means repeatedly applying a rule to improve an estimate. A rearranged equation can provide a rule x_(n+1) = f(x_n): start with the given x0 and substitute each result into the next step. If the estimates settle, their limiting value satisfies x = f(x). Some rearrangements diverge or oscillate, so a working rule must be checked rather than assumed. Keep full calculator precision during the process and round only the reported terms or final approximation.

Learning objectives

  • Use a starting value to generate successive estimates.
  • Recognise a fixed point and keep unrounded intermediate values.

Key ideas

The next estimate is calculated from the latest estimate.

xn+1=f(xn),x0=starting valuex_{n+1}=f(x_n),\qquad x_0=\text{starting value}

Keep full calculator values between iterations; round only the reported answers.

A fixed point satisfies this equation; iteration does not always converge.

x=f(x)x=f(x)

A sign change for a continuous function guarantees at least one root between the two inputs.

Worked examples

Example 1

Start at two and use the recurrence. Report the first three estimates to three decimal places.

xn+1=xn+4,x0=2x_{n+1}=\sqrt{x_n+4},\qquad x_0=2
x1=6=2.449489742…x_1=\sqrt6=2.449489742\ldots
x2=x1+4=2.539584561…x_2=\sqrt{x_1+4}=2.539584561\ldots
x3=x2+4=2.557261144…x_3=\sqrt{x_2+4}=2.557261144\ldots

Answer

x1≈2.449,x2≈2.540,x3≈2.557x_1\approx2.449,\quad x_2\approx2.540,\quad x_3\approx2.557

Example 2

Start at four and calculate the second estimate.

xn+1=0.5xn+3,x0=4⇒x1=5x_{n+1}=0.5x_n+3,\quad x_0=4\quad\Rightarrow\quad x_1=5

Answer

x2=0.5(5)+3=5.5x_2=0.5(5)+3=5.5

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Apply an iteration formula and give successive values to the required accuracy.

AO2 · Reasoning

Show that a root lies in an interval or explain whether an iteration appears to settle.

AO3 · Problem solving

Rearrange an equation into an iterative form and interpret an approximate root.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Use the full latest estimate as the next input, not its rounded display value. Check whether the values actually settle.

Try it yourself

Attempt each question before opening its answer.

Check 1: Use x_(n+1) = 0.5x_n + 3 with x0 = 4. Find x2.

Answer: 5.5

x1 = 5, then x2 = 0.5 × 5 + 3 = 5.5.

Check 2: What fixed point satisfies x = 0.5x + 3?

Answer: 6

Subtract 0.5x to get 0.5x = 3.

Explain it in your own words

  • Explain the meanings of x0, x1 and x2.
  • Show the first two substitutions for a given iteration rule.
  • Explain why intermediate estimates should not be rounded.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.