Sequences and series.
Use arithmetic, geometric and recurrence models, calculate finite and infinite sums, and justify formulas by induction.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Recurrence rules and explicit terms
Describe terms directly or generate them one step at a time, with a clear starting index.
- 02
Arithmetic sequences
Use constant differences to find terms, positions and missing sequence parameters.
- 03
Arithmetic sums
Derive arithmetic sums by pairing terms and apply them to totals and unknown term counts.
- 04
Geometric sequences
Recognise constant ratios and handle fractional ratios, alternating signs and index conventions.
- 05
Finite geometric sums
Derive finite geometric sums and apply them to alternating totals and correctly timed deposits.
- 06
Infinite geometric sums and convergence
Decide whether an infinite geometric sum exists and interpret its limiting value and remaining tail.
- 07
Recurrence models
Translate repeated proportional and fixed changes into recurrences, then interpret equilibrium and long-term behaviour.
- 08
Financial and other sequence applications
Choose arithmetic, geometric and recurrence models for finances and other repeated changes.
- 09
Introduction to proof by induction
Prove a statement for every allowed integer using a base case and an implication step.
- 10
Induction proofs of simple sums and arithmetic/geometric-series results
Prove finite arithmetic and geometric sum formulas, keeping parameters fixed and restrictions explicit.
Topic review
Revisit the key ideas, then choose a practice paper.