Iteration is a method for finding decimal solutions to equations that resist algebraic methods. You rearrange the equation into the form xₙ₊₁ = f(xₙ), pick a starting value, and feed each answer back in as the next input. Repeating this process homes in on the solution to any required degree of accuracy.
What does iteration mean in maths?
The word "iterate" simply means "repeat." In GCSE maths, iteration refers to a specific method where you:
- Express an equation in a rearranged form xₙ₊₁ = g(xₙ).
- Substitute a starting estimate x₀ to find x₁.
- Substitute x₁ to find x₂.
- Continue until consecutive values agree to the required number of decimal places.
The sequence x₀, x₁, x₂, … is called an iterative sequence, and when it settles down it converges to a root of the original equation. You do not need to prove it converges — at GCSE, the exam will always give you an iterative formula that works.
How do you show that a root lies in a given interval?
Before iterating, GCSE questions often ask you to show that a root lies between two integers, say between 2 and 3. You do this with the change-of-sign method:
- Define f(x) = (the equation rearranged so one side is 0).
- Calculate f(2) and f(3).
- If they have opposite signs, there must be a root between 2 and 3 (by the intermediate value theorem — you don't need to name it, just state the sign change).
Example: Show that x³ + x − 7 = 0 has a root between x = 1 and x = 2.
- f(1) = 1 + 1 − 7 = −5 (negative)
- f(2) = 8 + 2 − 7 = +3 (positive)
Sign changes from negative to positive, so there is a root in the interval (1, 2). ✓
How do you use an iterative formula step by step?
Worked example: The equation x³ + x = 7 can be written as xₙ₊₁ = (7 − xₙ)^(1/3) … actually let me use the correct rearrangement. It rearranges to:
$$x_{n+1} = \sqrt[3]{7 - x_n}$$
Starting with x₀ = 1.6, find x₁, x₂, x₃, and x₄, giving your answers to 4 decimal places.
| n | xₙ | Calculation | xₙ₊₁ |
|---|---|---|---|
| 0 | 1.6 | ∛(7 − 1.6) = ∛5.4 | 1.7520 |
| 1 | 1.7520 | ∛(7 − 1.7520) = ∛5.248 | 1.7358 |
| 2 | 1.7358 | ∛(7 − 1.7358) = ∛5.2642 | 1.7371 |
| 3 | 1.7371 | ∛(7 − 1.7371) = ∛5.2629 | 1.7370 |
| 4 | 1.7370 | ∛(7 − 1.7370) = ∛5.263 | 1.7370 |
The values have converged to x ≈ 1.737 to 3 decimal places. You would normally stop once two consecutive values agree to the number of decimal places required by the question.
How do you enter iterative calculations on a calculator?
Most scientific calculators allow you to store the answer and recall it. The efficient method is:
- Type your starting value and press = (or EXE on Casio).
- Type the iterative formula using ANS wherever xₙ appears.
- Press = repeatedly — each press gives the next iterate.
On a Casio CLASSWIZ: type 1.6 then =; then type (7 − ANS)^(1÷3) and press = repeatedly. This is much faster than retyping the formula each time and reduces the chance of keying errors.
How do you decide to what accuracy to state the answer?
The question will usually specify: "correct to 2 decimal places," or "give your answer to 3 significant figures." Continue iterating until two consecutive values agree when rounded to one more decimal place than required (to avoid rounding errors at the last digit).
Checking your answer: once you have a candidate root, verify it is correct to the required precision by substituting values either side. For example, if you claim the root is 1.737 to 3 d.p., check:
- f(1.7365) < 0 and f(1.7375) > 0 (or vice versa)
This confirms the root lies in the interval [1.7365, 1.7375), so it rounds to 1.737. ✓
What mistakes should you avoid?
The most common errors in GCSE iteration questions are:
- Using the wrong number of iterations. Always carry on until the values are stable to the required accuracy — one or two iterations is rarely enough.
- Rounding too early. Keep full calculator precision for each iterate; only round your final answer.
- Substituting back into the original equation instead of the iterative formula. They are different expressions; use xₙ₊₁ = g(xₙ) at every step.
- Sign errors in the change-of-sign check. Make sure you state the sign of f at each endpoint and say "sign change therefore root." Both parts must be present for full marks.
Frequently asked questions
What if the iterative sequence diverges instead of converging?
If the values get further from the root with each iteration, the sequence diverges and the formula does not work for that starting value (or at all). At GCSE you will always be given a formula that converges for the starting value provided, so if your values are growing — check your algebra and calculator input first, as the most likely cause is a keying error.
Do I need to know why iteration works?
No. At GCSE you apply the method without needing to prove convergence. The underlying idea (that you are finding a fixed point of the function g where g(x) = x corresponds to a root of the original equation) is A-level content.
Can I use any rearrangement of the equation?
The exam will give you the rearrangement. Different rearrangements of the same equation converge at different rates, and some diverge — but choosing the rearrangement is not part of GCSE. Your job is to apply the given formula correctly.
How many marks does an iteration question typically carry?
A full iteration question at GCSE Higher usually carries 5–6 marks split as: 1–2 for the change-of-sign showing the root exists, 3–4 for carrying out four or five iterations and stating the root to the required accuracy. Show all your iterates in a table — working shown line by line protects method marks even if you make an arithmetic slip.
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