A Level · Cambridge (CIE) · 9709

Interpret iteration notation and verify an equivalent fixed-point equation

Read subscripts as sequence positions, use each latest value in the given rule and check that a possible limit solves the original equation.

Section 3.6 · Numerical solution of equations

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

Substitute into an algebraic function and rearrange equations.
Recognise square-root and denominator domain restrictions.

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

Quick readiness check

What does x₂ mean in iteration notation?

What you will learn

Use x₀, x₁, … and xₙ₊₁ = F(xₙ) to generate a sequence of approximations.
Relate a convergent sequence's fixed point to the original equation, preserving rearrangement restrictions.

A subscript labels a term, not a power or multiplier

The supplied starting value x₀ is the seed. In xₙ₊₁ = F(xₙ), evaluate F at the current term to get the next one: x₁ = F(x₀), x₂ = F(x₁), and so on. The subscript n counts steps; x₂ is not x squared and xₙ is not n times x. Use each newly computed value as the next input, rather than substitute the original seed repeatedly.

For xₙ₊₁ = √(xₙ + 1) and x₀ = 1, the terms begin x₁ = √2 ≈ 1.414214, x₂ = √(1 + √2) ≈ 1.553774 and x₃ ≈ 1.598053. They approach the positive value (1 + √5)/2 ≈ 1.618034 in this example. Keep the full stored calculator value for the next step; the displayed decimals are for reporting, not replacements for the internal input.

A valid limit must satisfy both the rule and the original equation

If the sequence converges to r and F is continuous there, taking the limit in xₙ₊₁ = F(xₙ) gives r = F(r). This is a fixed point, seen graphically where y = F(x) meets y = x. Convergent iterates may approach from one side or alternate across the limit. Merely finding a fixed point does not prove that a chosen starting value will approach it; the actual iteration can fail.

The rearrangement x = √(x + 1) implies x² − x − 1 = 0, but also requires x ≥ 0. Squaring gives an additional negative quadratic root (1 − √5)/2, which does not satisfy the original square-root fixed-point equation. For the requested positive root, the given recurrence is appropriate. When rearranging, retain lost denominators, chosen root branches and log arguments, then substitute the candidate back into the original equation. A fixed point of a changed or undefined rule is not enough.

Pause and explain

Does the negative root of r² − r − 1 = 0 satisfy r = √(r + 1)?

Put the idea to work

Worked example

For xₙ₊₁ = √(xₙ + 1), x₀ = 1, calculate the first three new terms and identify the original equation solved by a positive limit.

Show the worked solution

x₁ = √2 ≈ 1.414214.
Use the latest value: x₂ = √(1 + √2) ≈ 1.553774 and x₃ ≈ 1.598053.
A continuous-rule limit satisfies r = √(r + 1), hence r² − r − 1 = 0 with r ≥ 0; its positive root is (1 + √5)/2.

Answer x₁ ≈ 1.414214, x₂ ≈ 1.553774, x₃ ≈ 1.598053; positive limit solves r² − r − 1 = 0 with r ≥ 0.

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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 xₙ₊₁ = (xₙ + 2)/3 with x₀ = 0, find x₁, x₂ and x₃ exactly, then solve the fixed-point equation.

Give me a hint

Use each latest fraction as the next input; solve r = (r + 2)/3 for the possible limit.

Compare my reasoning

The first new term is x₁ = 2/3.
Then x₂ = 8/9 and x₃ = 26/27.
The fixed point satisfies 3r = r + 2, giving r = 1; these terms move toward that value.

x₁ = 2/3, x₂ = 8/9, x₃ = 26/27; fixed point r = 1.

Look for these in your work

Advance one step using each latest term.
Relate a possible sequence limit to its fixed-point equation.

2 · Independent

A given rule for x³ + x − 1 = 0 is xₙ₊₁ = 1/(1 + xₙ²). Show the fixed-point equation is equivalent to the original equation and check its denominator.

Give me a hint

Multiply the fixed-point equation by 1 + r² only after confirming it cannot be zero for real r.

Compare my reasoning

At a fixed point, r = 1/(1 + r²).
For real r, 1 + r² > 0, so multiplication preserves the equation.
The result r + r³ = 1 is exactly r³ + r − 1 = 0; the fixed point is positive because the reciprocal is positive.

Equivalent fixed-point equation r³ + r − 1 = 0; denominator 1 + r² is always positive for real r.

Look for these in your work

Check the denominator before reversing the rearrangement.
Recover the original equation and retain any implied sign information.

3 · Transfer

A learner squares r = √(r + 1) and keeps both roots (1 ± √5)/2. Which candidate is valid for the original fixed-point equation and why?

Give me a hint

The original right side is the non-negative square root.

Compare my reasoning

Squaring gives r² − r − 1 = 0 with two algebraic candidates.
The negative candidate cannot equal a non-negative square root.
The positive candidate (1 + √5)/2 is non-negative and satisfies r² = r + 1, so it satisfies the original square-root equation.

Only (1 + √5)/2 is valid; the negative candidate fails the original non-negative branch.

Look for these in your work

Retain the original branch restriction after squaring.
Verify the accepted candidate in the unsquared equation.

Common mistakes

Subscripts count sequence positions and are not exponents.
Feed the latest unrounded term into the next step.
A rearrangement must keep its branch and domain restrictions before a fixed point counts as an original solution.

Recall without your notes

What can you explain now?

In xₙ₊₁ = F(xₙ), which input should be used to calculate x₂?

Compare with the explanation

x₁, so x₂ = F(x₁).

The rule advances one sequence position using the latest term.

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Make it stick

Explain what an iteration subscript means and how a continuous-rule limit becomes a fixed point. Give a rearrangement where checking the original branch removes an extra candidate.

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.