Sheet № 95 · Higher only · AQA · Edexcel · OCR
Iteration Method –
The iteration method is a Higher-tier technique for finding approximate solutions to equations that cannot be solved algebraically. By applying a formula repeatedly, each answer gets closer to the true solution. This guide focuses on applying the iterative formula, demonstrating rearrangements, and determining solutions to a given number
§Key definitions
Question:
Use the iterative formula xₙ₊₁ = (xₙ² + 3) / 5 with x₀ = 1 to find x₁, x₂ and x₃ to 4 decimal places.
Answer:
x₁ = 0.8000, x₂ = 0.7280, x₃ = 0.7060
Q1 (Higher):
Use xₙ₊₁ = ∛(2xₙ + 7) with x₀ = 2 to find x₁, x₂ and x₃ to 4 decimal places.
Q2 (Higher):
Show that x³ − 6x + 2 = 0 can be rearranged to x = (x³ + 2) / 6.
Q3 (Higher):
Use xₙ₊₁ = ∛(6xₙ − 2) with x₀ = 2 to find the solution of x³ − 6x + 2 = 0 correct to 2 decimal places.
§Formulas to memorise
xₙ₊₁ = f(xₙ) — the iterative formula
The solution is found when consecutive iterations agree to the required decimal places
Write down — the iterative formula and the starting value x₀.
Substitute — x₀ into the formula to calculate x₁. Write the full calculator display.
Substitute — x₁ to find x₂. Use the unrounded value from the previous step.
Repeat — until the question is answered — either a set number of iterations or until values agree to the required decimal places.
Never round — intermediate values. Only round your final answer.
State — the solution clearly, to the accuracy requested.
Worked example
Use the iterative formula xₙ₊₁ = (xₙ² + 3) / 5 with x₀ = 1 to find x₁, x₂ and x₃ to 4 decimal places.
Working:
⚠ Common mistakes
- ✗Rounding intermediate values. Using a rounded value in the next step compounds errors. Always use the full unrounded calculator display and only round the final answer.
- ✗Not showing enough iterations. You must continue until two consecutive values agree to the required number of decimal places. Stopping early loses accuracy marks.
- ✗Substituting incorrectly into cube roots. Be careful with brackets on your calculator. ∛(4x + 1) is different from ∛4 × x + 1.
✦ Exam tips
- →Use the ANS button on your calculator: type the formula once using ANS in place of xₙ, then press = repeatedly to generate successive values.
- →Write every iteration value to at least 6 decimal places to demonstrate you are not rounding early.
- →If asked to show a rearrangement leads to the iterative formula, start from the original equation and work algebraically — do not work backwards.