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Algebra · Higher

Iteration

Iteration is a numerical method for finding approximate solutions to equations by repeatedly applying a rearrangement formula, starting with an initial estimate.

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Key facts to remember

  • 1An iterative formula has the form xₙ₊₁ = f(xₙ) — each new value is calculated from the previous one.
  • 2Start with an initial value x₀ (usually given in the question).
  • 3Repeat the process until the answer converges (successive values agree to the required d.p.).
  • 4Iteration finds approximate roots, not exact ones.
  • 5The formula must be rearranged from the original equation.
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Worked examples

Example 1

Use the iterative formula xₙ₊₁ = √(5 + xₙ) with x₀ = 3 to find x₃ to 3 d.p.

Working

  1. x₁ = √(5 + 3) = √8 = 2.8284…
  2. x₂ = √(5 + 2.8284) = √7.8284 = 2.7979…
  3. x₃ = √(5 + 2.7979) = √7.7979 = 2.7925…
  4. x₃ ≈ 2.793
Answerx₃ ≈ 2.793
Example 2

Show that x³ − x − 7 = 0 has a root between x = 2 and x = 3.

Working

  1. Let f(x) = x³ − x − 7
  2. f(2) = 8 − 2 − 7 = −1 (negative)
  3. f(3) = 27 − 3 − 7 = 17 (positive)
  4. Sign change → root lies between x = 2 and x = 3
AnswerSign change between f(2) = −1 and f(3) = 17 confirms a root exists in the interval.
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Common mistakes

✗Using xₙ in the formula before computing xₙ₊₁ — always compute step by step.
✗Rounding intermediate values, which compounds errors — keep full calculator accuracy.
✗Not identifying a sign change correctly when showing a root exists.
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Exam tips

✓Store each iteration in your calculator memory to avoid rounding errors.
✓Show all iterations clearly and state the final answer to the required accuracy.

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