A Level · Cambridge (CIE) · 9709

Run given iterations with retained precision and original-equation checks

Generate the requested terms, retain calculator precision and support the final rounded root with evidence from the original continuous equation.

Pure Mathematics 2 & 3 · Numerical solution of equations · 2.6 / 3.6

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Paper 1 graph and algebra knowledge is assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Shared sections 2.6 and 3.6 include locating roots graphically or by sign changes, notation for convergent approximation sequences, and using a given iteration or rearrangement to a prescribed accuracy. An iteration may fail to converge. Knowledge of the general condition for convergence is not required here. Check the original equation, domains and requested rounding. Written practice is self-checked.

Before you begin

Read iteration subscripts and recover the original fixed-point equation.
Retain function precision and use a valid root bracket for rounding.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which value should be carried into the next calculator iteration?

What you will learn

Use a given fixed-point rule or given rearrangement to approximate a root.
Check domain validity, original-equation signs and the requested accuracy before reporting the result.

The solid blue graph is F(x) = 1/(1 + x²) and the dashed grey line is y = x. Their intersection P = (r, r), where r is approximately 0.682327804, is a fixed point satisfying r³ + r − 1 = 0. The red stepping path shows four iterations from x₀ = 0.5: move vertically to the function, then horizontally to y = x to make the output the next input. The values 0.8, 0.609756098, 0.728967910 and 0.652999725 alternate around the root. The displayed steps illustrate the process; original-equation checks support the reported accuracy. Both axes use equal unit lengths.

Use the formula as given and retain working digits

For x³ + x − 1 = 0, use the given rule xₙ₊₁ = 1/(1 + xₙ²) with x₀ = 0.5. The terms begin x₁ = 0.8, x₂ ≈ 0.609756098, x₃ ≈ 0.728967910, x₄ ≈ 0.652999725 and x₅ ≈ 0.701061373. They alternate across a limiting value near 0.682327804. The denominator is positive for every real input, so the rule does not encounter a reciprocal pole.

Keep the full stored term in the next calculation, even when a table displays only a few digits. Rounding each step changes the numerical sequence and can produce a false fixed value or cycle. Label a requested table accurately: if x₀ is the starting value, the first newly generated term is x₁. Use the number of terms, starting value and rearrangement that the question specifies; a different rule can behave differently.

Validate the approximation in the original equation

For f(x) = x³ + x − 1, f(0.6823) = −0.000066636233 and f(0.6824) = 0.000173044224. Continuity therefore brackets a root between these inputs. Both endpoints and the full interval round to 0.682 at three decimal places, supporting the final report. The iteration supplies approximations; the original sign bracket gives separate evidence for the rounding claim. The polynomial is increasing for all real x because 3x² + 1 > 0, so this root is unique.

Other supplied rules may involve square roots or logarithms. For xₙ₊₁ = √(3 − xₙ), every actual input must satisfy xₙ ≤ 3 and every output is non-negative. For xₙ₊₁ = ln(3 − xₙ), every actual input must satisfy xₙ < 3. Check those restrictions at each step and verify the final candidate in its original equation, since a squared rearrangement or a changed log domain can alter which solutions are admissible. These examples do not require the general convergence condition as a syllabus topic.

Pause and explain

Why check f(0.6823) and f(0.6824) in the original cubic?

Put the idea to work

Worked example

Use xₙ₊₁ = 1/(1 + xₙ²), x₀ = 0.5, to approximate the root of x³ + x − 1 = 0 to three decimal places, with an original-equation check.

Show the worked solution

Generate successive terms 0.8, 0.609756098, 0.728967910, 0.652999725, … while retaining full stored precision.
Continue toward approximately 0.682327804; every denominator remains positive.
Original values at 0.6823 and 0.6824 have opposite signs, and the entire bracket rounds to 0.682 at three decimal places.

Answer Root approximately 0.682 to three decimal places, supported by the bracket (0.6823, 0.6824).

Try a topic test

Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

For x³ − x − 2 = 0, use the given rule xₙ₊₁ = ∛(xₙ + 2), x₀ = 1.5. Find the first three new terms and a root to three decimal places, checking the bracket (1.5213, 1.5214).

Give me a hint

Use real cube roots and retain each unrounded term in the next step.

Compare my reasoning

The first three terms are approximately 1.518294486, 1.520935263 and 1.521315698.
Further terms approach 1.521379707; the real cube-root rearrangement recovers the original cubic.
Original values at 1.5213 and 1.5214 are approximately −0.000473731403 and 0.000120620344. The continuous bracket rounds throughout to 1.521.

x₁ ≈ 1.518294486, x₂ ≈ 1.520935263, x₃ ≈ 1.521315698; root approximately 1.521 to three decimal places.

Look for these in your work

Use the given seed and latest full-precision term.
Recover the original equation and check signs within a single rounding band.

2 · Independent

Use xₙ₊₁ = √(3 − xₙ), x₀ = 1, to approximate the positive root of x² + x − 3 = 0 to two decimal places. Check the square-root branch and bracket (1.3027, 1.3028).

Give me a hint

Keep the non-negative output restriction when relating the recurrence to the quadratic.

Compare my reasoning

The sequence begins 1.414213562, 1.259280127, 1.319363435, … and approaches approximately 1.302775638.
Every calculated input remains within the square-root domain; its fixed point is non-negative, so the negative quadratic root is not a valid square-root fixed point.
The original values at 1.3027 and 1.3028 are −0.00027271 and 0.00008784. This bracket lies wholly in the two-decimal rounding band for 1.30.

Positive root approximately 1.30 to two decimal places; the non-negative square-root branch must be retained.

Look for these in your work

Check the actual sequence's real domain and chosen positive branch.
Support the requested two-decimal answer with original-equation signs.

3 · Transfer

For the supplied index R(t) = eᵗ + t on t ≥ 0 hours, use tₙ₊₁ = ln(3 − tₙ), t₀ = 1, to find when R reaches 3 to three decimal places. Check the original bracket (0.7920, 0.7921).

Give me a hint

The recurrence requires 3 − tₙ > 0; its fixed point must also satisfy eᵗ + t = 3.

Compare my reasoning

The terms begin approximately 0.693147181, 0.835884180, 0.772011881, … and approach 0.792059968, with positive log arguments throughout the calculated sequence.
For f(t) = eᵗ + t − 3, the endpoint values are approximately −0.000192371159 and 0.000128420643.
Continuity and the common rounding band give t ≈ 0.792 hours; this is an approximate threshold in the supplied model.

Threshold time approximately 0.792 hours to three decimal places, with a valid logarithmic iteration and original-equation bracket.

Look for these in your work

Use the given time recurrence within its positive log-argument domain.
Verify the model equation and report the requested accuracy with time units.

Common mistakes

Retain more digits internally than are displayed or finally requested.
Check every iterate's square-root, reciprocal or logarithm domain.
Final digit stability should be checked against the original equation and requested rounding.

Recall without your notes

What can you explain now?

For xₙ₊₁ = 1/(1 + xₙ²) with x₀ = 0.5, what are x₁ and x₂?

Compare with the explanation

x₁ = 0.8; x₂ = 25/41 ≈ 0.609756098.

The second step uses the newly obtained 0.8, not the original seed again.

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Make it stick

Run a given recurrence without premature rounding. Explain how an original-domain check and a narrow sign-change bracket support the final root report.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.