Iteration
Iteration is a key number topic at GCSE Maths. This Higher worksheet gives you exam-style questions on using iteration to solve equations, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Substitute each answer back into the formula to get closer to the solution.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.
Topic overview
Iteration finds approximate solutions to equations that cannot be solved exactly by algebra. You start with an estimate, feed it into a formula, and use the result as the next estimate.
Each pass gets closer to the true solution, a process called converging. In the exam you are given the iterative formula, so the skill is applying it accurately rather than deriving it.
The practical detail is using the answer key on the calculator. Type the first estimate, then enter the formula using the answer key where \(x_n\) appears, and press equals repeatedly. Each press produces the next iteration, which is far quicker and more accurate than retyping rounded values.
Revision notes
Applying the formula
The notation \(x_{n+1} = f(x_n)\) means the next value comes from substituting the current one.
Given \(x_{n+1} = \sqrt{3x_n + 2}\) with \(x_0 = 2\): \(x_1 = \sqrt{8} = 2.828\ldots\), then \(x_2 = \sqrt{3(2.828) + 2} = 3.238\ldots\), and so on.
Using the calculator efficiently
Type the starting value and press equals. Then enter the formula using the answer key in place of \(x_n\), and press equals repeatedly.
Each press gives the next iteration with full accuracy retained, which matters because rounding between steps changes later values.
Recognising convergence
The values settle towards a fixed number. When successive iterations agree to the required accuracy, you have the solution.
Round only at the end, and state the answer to the accuracy the question specifies, usually three decimal places.
Key points
- Iteration finds approximate solutions by repetition.
- \(x_{n+1} = f(x_n)\) means substitute the current value to get the next.
- The formula is given in the question.
- Use the answer key to keep full accuracy.
- Values converge towards the solution.
- Round only at the final step.
Worked examples
Example 1
Using \(x_{n+1} = \sqrt{2x_n + 3}\) with \(x_0 = 3\), find \(x_1\).
Working
Example 2
Using \(x_{n+1} = \dfrac{5}{x_n + 1}\) with \(x_0 = 2\), find \(x_1\) and \(x_2\).
Working
Example 3
Explain why the answer key should be used rather than retyping values.
Working
Common mistakes
Rounding between iterations.
Each rounded value feeds into the next, so errors compound. Use the answer key.
Substituting into the wrong side.
The current value goes into the right-hand side, and the result becomes the next value.
Stopping too early.
Continue until successive values agree to the required accuracy.
Not giving the answer to the stated accuracy.
If three decimal places are asked for, give exactly that.
Exam tips
- Write each iteration on its own line with its subscript.
- Use the answer key on the calculator rather than retyping.
- Keep full accuracy until the final rounding.
- State the solution to the accuracy the question requires.
Key terms
- Iteration
- Repeating a process to get closer to a solution.
- Iterative formula
- A formula giving the next value from the current one.
- Converge
- To settle towards a particular value.
- Subscript
- The small number labelling each value in a sequence.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.