lesson
An update rule does not guarantee convergence
1 · Learn
Iteration repeatedly applies an update rule to a starting value. Record enough precision to see the pattern and distinguish an apparent approach to a solution from proof of convergence. Different rearrangements can behave differently.
A fixed point satisfies x = F(x), but a sequence may oscillate or diverge instead of approaching it. Always state the starting value and rule.
2 · Worked example
With x(next) = (x + 6)/2 and start 0, values are 3, 4.5, 5.25. The fixed point solves x = (x + 6)/2, giving 6.
3 · Your turn
Using the same rule, what follows 5.25?
Check your answer
5.625.
4 · Apply your learning
Compare with the rule x(next) = 6 - x starting at 0, which alternates 0 and 6 rather than converging to its fixed point 3.
Adult guidance and safety
This sequence includes extension mathematics and assumes secure earlier work. Check prerequisite understanding and the learner's readiness before assigning extension tasks. Require derivations, accurate diagrams and varied practice; a correct short example is not full assessment evidence. Exam-board and tier-specific pathways are outside this edition.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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