Sequences and Series
Work with explicit and recursive sequences, arithmetic and geometric sums and positive-integer binomial expansions.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Move between term-to-term rules, nth terms and recurrence relations.
Exam cue: Generate several terms before generalising a pattern.
Concept 2
Use finite arithmetic and geometric sums and test convergence before an infinite sum.
Exam cue: Identify the first term, common difference or ratio explicitly.
Concept 3
Expand binomials with correct coefficients and matching powers.
Exam cue: For an infinite geometric series, test the magnitude of the ratio first.
Risk pitfalls and guardrails
Using a finite-sum formula for an infinite series.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Assuming an observed pattern is proved by a few cases.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Dropping binomial coefficients or mismatching powers.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Arithmetic step
Consecutive terms differ by one constant amount.
Geometric ratio
Consecutive non-zero terms share one multiplier.
Finite geometric sum
Track the first term, ratio and number of terms.
Convergence gate
An infinite geometric sum needs |r| < 1.
Recurrence
Each term is generated from an earlier term and the rule.
Binomial balance
Coefficients pair with powers moving in opposite directions.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
An arithmetic sequence has u₄=11 and u₉=31. What is u₁+u₁₀?
The sum of the first n odd positive integers is 169. What is the next odd integer?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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