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
x₁ = √(5 + 3) = √8 = 2.8284…
x₂ = √(5 + 2.8284) = √7.8284 = 2.7979…
x₃ = √(5 + 2.7979) = √7.7979 = 2.7925…
x₃ ≈ 2.793
Answerx₃ ≈ 2.793
Example 2
Show that x³ − x − 7 = 0 has a root between x = 2 and x = 3.
Working
Let f(x) = x³ − x − 7
f(2) = 8 − 2 − 7 = −1 (negative)
f(3) = 27 − 3 − 7 = 17 (positive)
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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