AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Solving equations and inequalities: find approximate solutions to equations numerically using iteration (Higher tier only)
Iteration finds approximate solutions to equations that cannot be solved exactly.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Iteration finds approximate solutions to equations that cannot be solved exactly. Rearrange the equation into the form x = g(x), choose a starting value x₀, and repeatedly apply the formula xₙ₊₁ = g(xₙ). Each result is substituted back until successive values settle to a required accuracy. For example, to solve x³ + x = 20, rearrange to x = ∛(20 − x). Starting at x₀ = 2 gives x₁ = ∛18 ≈ 2.6207, x₂ ≈ 2.5903, x₃ ≈ 2.5918, converging near 2.592. A good starting value and a convergent rearrangement are essential, as some rearrangements diverge. You must show enough iterations to justify the accuracy of your answer and interpret the result in context.
What earns the marks
- Rearranges the given equation correctly into the iterative form x = g(x), for example x = ∛(20 − x) from x³ + x = 20.
- Uses a sensible starting value x₀ and substitutes it correctly into the iteration formula to obtain x₁.
- Repeats the iteration accurately, showing successive values, until two consecutive values agree to the required degree of accuracy.
- States the approximate solution to the required accuracy, for example x ≈ 2.59 to 2 decimal places, and shows sufficient iterations to justify it.
- Recognises when an iteration diverges or oscillates and adjusts the rearrangement or starting value accordingly.
Common mistakes and how to put them right
Substituting into the original equation instead of the rearranged iterative formula.
Put it right: Apply xₙ₊₁ = g(xₙ) at each step.
Rounding intermediate values too early, which prevents convergence.
Put it right: Keep full calculator accuracy and round only the final answer.
Assuming any rearrangement will converge.
Put it right: Test the iteration and, if values diverge, choose a different rearrangement or starting value.
Stopping after one or two iterations without evidence of convergence.
Put it right: Continue until successive values agree to the required accuracy.
Exam tips
- Always write down your rearrangement and starting value before iterating, so the method is clear.
- Show at least three or four iterations in a table, keeping full calculator display values until the final rounding.
- Check convergence by comparing consecutive values; if they are not settling, reconsider your rearrangement.
Try the lesson
The start of the Solving equations and inequalities: find approximate solutions to equations numerically using iteration (Higher tier only) lesson, made from checked teaching notes for this specification point.
Have a go first
A sequence is defined by the iteration formula xₙ₊₁ = (xₙ + 7) / 2 with starting value x₁ = 5. Work out the value of x₃.
Listen
Iteration finds approximate solutions to equations that cannot be solved exactly. Rearrange the equation into the form x = g(x), choose a starting value x₀, and repeatedly apply the formula xₙ₊₁ = g(xₙ). Each result is substituted back until successive values settle to a required accuracy. For example, to solve x³ + x = 20, rearrange to x = ∛(20 − x). Starting at x₀ = 2 gives x₁ = ∛18 ≈ 2.6207, x₂ ≈ 2.5903, x₃ ≈ 2.5918, converging near 2.592. A good starting value and a convergent rearrangement are essential, as some rearrangements diverge. You must show enough iterations to justify the accuracy of your answer and interpret the result in context. A strong answer earns marks for these points. Rearranges the given equation correctly into the iterative form x = g(x), for example x = ∛(20 − x) from x³ + x = 20. Uses a sensible starting value x₀ and substitutes it correctly into the iteration formula to obtain x₁. Repeats the iteration accurately, showing successive values, until two consecutive values agree to the required degree of accuracy. States the approximate solution to the required accuracy, for example x ≈ 2.59 to 2 decimal places, and shows sufficient iterations to justify it. A common slip: Substituting into the original equation instead of the rearranged iterative formula. Instead, apply xₙ₊₁ = g(xₙ) at each step. Exam tip: Always write down your rearrangement and starting value before iterating, so the method is clear.
Key words
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Iteration
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Convergence
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Starting value
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Divergence
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See every definition
- Iteration
- A method that finds approximate solutions to equations that cannot be solved exactly by repeatedly applying a formula.
- Convergence
- Successive values settling to a required accuracy, which a good starting value and a convergent rearrangement are essential for.
- Starting value
- The value x₀ chosen before repeatedly applying the formula xₙ₊₁ = g(xₙ).
- Divergence
- What happens when some rearrangements do not settle, so the rearrangement or starting value must be adjusted.
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