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 you understand the setup, and it is a great opportunity to pick up marks with careful calculator work. This guide explains how iteration works, walks through examples, and highlights the common pitfalls.

What Is Iteration?

Iteration is a method for finding approximate solutions to equations that cannot be solved exactly using algebra. You start with an initial value (called x₀) and substitute it into an iterative formula to generate successive approximations x₁, x₂, x₃, and so on.

Each new value is found by substituting the previous one into the formula. As you repeat the process, the values converge (settle down) towards the solution.

How iterative formulas are formed

An iterative formula is created by rearranging an equation into the form x = g(x). For example:

Starting equation: x³ + 2x = 7

Rearrange: x³ = 7 − 2x → x = ∛(7 − 2x)

So the iterative formula is:

When iteration converges

Iteration converges when successive values get closer and closer together, eventually agreeing to a specified number of decimal places. This stable value is the approximate solution to the original equation.

Step-by-Step Method

1. Write down the iterative formula and the starting value x₀.
2. Substitute x₀ into the formula to find x₁. Write down the full calculator display.
3. Substitute x₁ into the formula to find x₂. Again, write down the full value.
4. Repeat until the required number of iterations is complete, or until values agree to the required number of decimal places.
5. Round your final answer only at the end — use the full unrounded value at each step.
6. 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.

Worked Example 1 — Using an Iterative Formula

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...

x₃ = 2.401 (to 3 d.p.)

Notice how the values are settling down — they are converging towards the solution of x³ = 5x + 2.

Worked Example 2 — Finding a Solution to a Given Accuracy

Question: The equation x³ − 3x − 5 = 0 can be rearranged to give x = ∛(3x + 5). Using x₀ = 2, find the solution correct to 2 decimal places.

Iterations:

x₁ = ∛(3(2) + 5) = ∛11 = 2.223980...

x₂ = ∛(3(2.223980...) + 5) = ∛(11.67194...) = 2.264784...

x₃ = ∛(3(2.264784...) + 5) = ∛(11.79435...) = 2.277533...

x₄ = ∛(3(2.277533...) + 5) = ∛(11.83260...) = 2.279495...

x₅ = ∛(3(2.279495...) + 5) = ∛(11.83849...) = 2.279876...

x₄ and x₅ both round to 2.28 (to 2 d.p.).

Since two consecutive iterations agree to 2 decimal places, the solution is x = 2.28 (to 2 d.p.).

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

1. 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.
2. Write every iteration value to at least 6 decimal places. This shows the examiner you are not rounding early and earns method marks.
3. 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.
4. 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.

Practice Questions

Question 1: Use xₙ₊₁ = (xₙ³ + 4) / 7 with x₀ = 1. Find x₁, x₂ and x₃.

Question 2: Show that x³ + 5x = 8 can be rearranged to give x = (8 − x³) / 5.

Question 3: The iterative formula xₙ₊₁ = √(10 − 3xₙ) with x₀ = 2 converges to a solution of x² + 3x = 10. Find this solution correct to 1 decimal place.

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Related Topics

• Solving Quadratic Equations
• Rearranging Formulae
• Algebraic Proof
• Functions and Function Notation

Summary

• Iteration uses a formula xₙ₊₁ = g(xₙ) to find approximate solutions by repeated substitution.
• Start with x₀ and substitute into the formula to find x₁, then x₂, and so on.
• Use the ANS button on your calculator for speed and accuracy.
• Never round intermediate values — only round the final answer.
• The solution is found when two consecutive iterations agree to the required number of decimal places.
• Iterative formulas are formed by rearranging the original equation into the form x = g(x).
• Always show your working clearly, writing each iteration value to at least 6 decimal places.