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
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.
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
Iteration Process
Put xₙ into formula to get xₙ₊₁. Repeat until values converge (settle down).
Sum Formula
S_n = a(r^n - 1)/(r - 1) where r is common ratio