Knowledge Organiser: Iteration
Part of Iteration · GCSE GCSE Mathematics revision
This topic summary covers Knowledge Organiser: Iteration within Iteration for GCSE Mathematics. Revise Iteration in Algebra for GCSE Mathematics with 9 exam-style questions and 3 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 4 of 4 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 4 of 4
Practice
9 questions
Recall
3 flashcards
Knowledge Organiser: Iteration
Key Terms
- Iteration: Repeatedly applying a formula using each output as the next input
- Iteration formula: A formula of the form xₙ₊₁ = f(xₙ)
- Starting value x₀: The initial value given to start the process
- Convergence: When successive values get closer and closer to a fixed answer
- Fixed point: The solution — the value x where f(x) = x
Must-Know Facts
- Substitute the previous x value into the formula to get the next one
- Keep full decimal precision in each step — round only the final answer
- Continue until two consecutive values agree to the required accuracy
- Not all iteration formulas converge — they may diverge (move away)
- The converged value is a solution to the original equation
- Questions often ask for x₃ (applying the formula 3 times from x₀)
Key Methods
- Given x₀: calculate x₁ = f(x₀), then x₂ = f(x₁), then x₃ = f(x₂), etc.
- To find solution to n d.p.: iterate until two consecutive values round to the same value
- Rearrange equation to form xₙ₊₁ = f(xₙ) before iterating
- To verify a solution: substitute back into the original equation
Key Formulas
- Iteration formula: xₙ₊₁ = f(xₙ)
- Change of sign: if f(a) and f(b) have opposite signs, a root lies between a and b
- Verify to n d.p.: check f(x − 0.5 × 10⁻ⁿ) and f(x + 0.5 × 10⁻ⁿ) have opposite signs
Common Mistakes
- Using wrong starting value: Use the x₀ given in the question — a different starting value may converge to a different root
- Stopping too early: Iterate until two consecutive values agree to the required number of decimal places
- Rounding intermediate steps: Keep full calculator accuracy for each iteration — only round the final answer
- Verifying the answer: Show a change of sign in a small interval around your answer to confirm it is correct to the stated accuracy
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Practice Questions for Iteration
What is the purpose of using an iterative formula in mathematics?
A student uses the iterative formula xₙ₊₁ = xₙ² − 2 with x₀ = 0.5 and obtains the sequence 0.5, −1.75, 1.0625, −0.871, −1.241, ... Explain what is happening.
Quick Recall Flashcards
9 questions on Iteration — practise free
Instant marking, adaptive difficulty, and 3 spaced repetition flashcards. Free until your GCSEs.
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