Sheet № 32 · Higher only · AQA · Edexcel · OCR
Iteration
Iteration is a Higher-tier topic that appears on AQA, Edexcel and OCR GCSE Maths papers, typically worth three to four marks. It involves using a formula repeatedly to home in on the solution to an equation. Each time you substitute your answer back into the formula, you get a more accurate approximation. The process is mechanical once yo
§Key definitions
Calculator tip:
Use the ANS button on your calculator. Type the formula using ANS in place of xₙ, then press = repeatedly. Each press gives the next iteration.
Question:
Use the iterative formula xₙ₊₁ = ∛(5xₙ + 2) with x₀ = 2 to find x₁, x₂ and x₃. Give x₃ to 3 decimal places.
Step 1:
x₁ = ∛(5(2) + 2) = ∛12 = 2.289428...
Step 2:
x₂ = ∛(5(2.289428...) + 2) = ∛(13.44714...) = 2.374577...
Step 3:
x₃ = ∛(5(2.374577...) + 2) = ∛(13.87289...) = 2.401169...
§Formulas to memorise
xₙ₊₁ = g(xₙ)
xₙ₊₁ = ∛(7 − 2xₙ)
Write down the iterative formula — and the starting value x₀.
Substitute x₀ — into the formula to find x₁. Write down the full calculator display.
Substitute x₁ — into the formula to find x₂. Again, write down the full value.
Repeat — until the required number of iterations is complete, or until values agree to the required number of decimal places.
Round — your final answer only at the end — use the full unrounded value at each step.
State the solution — to the required degree of accuracy.
Calculator tip:: Use the ANS button on your calculator. Type the formula using ANS in place of xₙ, then press = repeatedly. Each press gives the next iteration.
Question:: Use the iterative formula xₙ₊₁ = ∛(5xₙ + 2) with x₀ = 2 to find x₁, x₂ and x₃. Give x₃ to 3 decimal places.
Step 1:: x₁ = ∛(5(2) + 2) = ∛12 = 2.289428...
Step 2:: x₂ = ∛(5(2.289428...) + 2) = ∛(13.44714...) = 2.374577...
Worked example
Use the iterative formula xₙ₊₁ = ∛(5xₙ + 2) with x₀ = 2 to find x₁, x₂ and x₃. Give x₃ to 3 decimal places.
Step 1: x₁ = ∛(5(2) + 2) = ∛12 = 2.289428...
⚠ Common mistakes
- ✗Rounding too early. Always use the full unrounded value from your calculator for the next iteration. Only round the final answer. Premature rounding introduces errors that compound through each step.
- ✗Not showing enough iterations. If the question asks you to find the solution to a given number of decimal places, you must show enough iterations for two consecutive values to agree to that accuracy.
- ✗Substituting incorrectly into the formula. Take care with the order of operations. Write out each substitution step clearly and use brackets on your calculator.
- ✗Confusing xₙ and xₙ₊₁. xₙ₊₁ is the new value — the output. xₙ is the value you put in.
- ✗Not verifying the formula. If asked to show that an equation rearranges to a given iterative formula, you must show clear algebraic steps — not just state the result.
✦ Exam tips
- →Use the ANS button on your calculator for efficiency. Type the formula once using ANS, then press = repeatedly to generate successive iterations. This avoids re-typing and reduces errors.
- →Write every iteration value to at least 6 decimal places. This shows the examiner you are not rounding early and earns method marks.
- →On AQA papers, you may be asked to "show that x³ − 3x − 5 = 0 can be rearranged to x = ∛(3x + 5)". Start from the equation and rearrange step by step — do not work backwards from the answer.
- →If the question asks for a solution to n decimal places, you need two consecutive iterations that agree when rounded to n d.p. State this explicitly in your answer.