Iterative processes
Iteration is an Edexcel 1MA1 Higher signature topic — it appears on virtually every Higher Paper 3 (calculator). It is rarely tested by other boards at this level.
What is iteration?
Iteration finds approximate solutions to equations that cannot be solved algebraically by repeatedly substituting values into a recurrence relation until the answer converges to a stable value.
A recurrence relation has the form xₙ₊₁ = f(xₙ).
Starting from an initial estimate x₀, you compute: x₁ = f(x₀) x₂ = f(x₁) x₃ = f(x₂) … and so on, until values stop changing to the required degree of accuracy.
✦Worked example— Example
The equation x³ + 2x − 5 = 0 has a solution near x = 1.3.
Rearrange to: x = (5 − 2x)/x² → wait, Edexcel gives you the formula; you do NOT need to derive it.
Suppose the formula given is xₙ₊₁ = (5 − 2xₙ) / xₙ².
With x₀ = 1.3: x₁ = (5 − 2.6) / 1.69 = 2.4 / 1.69 ≈ 1.4201 x₂ = (5 − 2.8402) / 2.0171 ≈ 1.0706 …
Converging sequences home in on a root. Diverging sequences move away — this means the formula is unsuitable for that starting value.
Using a Casio calculator efficiently
On Paper 3, set x = 1.3 [EXE], then type the formula referencing Ans: (5 − 2Ans) / Ans² [EXE] [EXE] [EXE] … Each press of EXE computes the next iterate. Stop when the required d.p. stabilises.
Standard Edexcel question structure
- "Show that the equation f(x) = 0 has a root between a and b." — Evaluate f(a) and f(b); show they have opposite signs (sign change → root by IVT).
- "Use the iterative formula xₙ₊₁ = g(xₙ) with x₀ = k to find x₁, x₂, x₃ to 3 d.p." — Show all working.
- "Write down the solution to f(x) = 0 to an appropriate degree of accuracy." — When consecutive iterates agree to the required d.p.
- "Verify your answer." — Substitute back into f(x); show the sign change confirms the root lies in a narrow interval.
⚠Common mistakes
- Premature rounding: carry full calculator precision between steps; only round the final answer.
- Wrong starting value: use the x₀ given in the question.
- Assuming convergence means the answer is exact: iterative methods give approximations.
- Not verifying: the final mark often requires a sign-change check.
- Confusing xₙ and xₙ₊₁: xₙ₊₁ is the OUTPUT; xₙ is the INPUT at each step.
Edexcel exam tip
The iteration formula is always given to you — you never derive it. The marks are for: correct substitution, correct iteration to specified d.p., and the verification (sign-change) argument.
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