What is important to know about Iteration?

An iterative formula has the form xₙ₊₁ = f(xₙ) — each new value is calculated from the previous one.

What is important to know about Iteration?

Start with an initial value x₀ (usually given in the question).

What is important to know about Iteration?

Repeat the process until the answer converges (successive values agree to the required d.p.).

📐

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.

🔑

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.

✍️

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.

⚠️

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.

🎯

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.

Ready to test yourself on Iteration?

Get AI-marked practice questions on exactly this subtopic.

Practice this topic →

← All topics Dashboard

▶️ Watch on YouTube

Free video lessons

Click a topic to search

▶iteration GCSE Higher maths ▶iterative formula GCSE maths ▶numerical methods GCSE Higher ▶iteration roots equations GCSE

Opens YouTube — pick any free GCSE video.