Iteration

Approaching an equation’s solution step by step

Iteration uses a repeated calculation to produce increasingly accurate approximations. GCSE Higher and IGCSE questions normally provide a starting value and a formula for generating the next value. The method builds on substitution, equations and calculator skills.

Key knowledge and method

Start with the given x-value, substitute it into the right-hand side of the iterative formula, store the result, and use that result as the next input. Continue until successive values agree to the required accuracy. Do not round during intermediate steps. Review rearranging formulae and the meaning of a formula

Worked examples

Example 1: Use xₙ₊₁ = (xₙ + 6) ÷ 3 with x₁ = 3. Then x₂ = 3, so the value is already stable. Example 2: Use xₙ₊₁ = √(10 - xₙ) with x₁ = 2. Then x₂ ≈ 2.828, x₃ ≈ 2.678 and x₄ ≈ 2.706. Example 3: If consecutive values are 2.70154 and 2.70156, the root is approximately 2.702 to three decimal places.

Common mistakes and exam advice

Students may reuse the starting value instead of the latest result, enter brackets incorrectly or round too early. Write a clear table of x₁, x₂, x₃ and so on, and keep the full calculator display until the final answer. Knowledge of quadratic equations helps explain why an approximate root is needed.

Practice exercises and answers

Exercise 1: xₙ₊₁ = (xₙ + 8) ÷ 2, x₁ = 4. Find x₂. Answer: 6. Exercise 2: Use the same formula to find x₃. Answer: 7. Exercise 3: xₙ₊₁ = √(12 - xₙ), x₁ = 3. Find x₂ and x₃. Answer: x₂ = 3 and x₃ = 3, so the process has converged to 3.

Related learning across the website

Strengthen iteration through Algebra Lessons, GCSE algebra help, linear equations, quadratic equation lessons, sequences, equations, variables, powers and standard form

Benefits of one-to-one maths tutoring

One-to-one lessons with maths tutor Ryan Harvey can improve formula entry, calculator workflow and understanding of convergence through carefully graded practice.

Contact and lesson enquiry

Students and parents can contact MasterMaths Tutoring for personalised iteration support.