What to look for in a recurrence
If the recurrence is affine, shift by its fixed point. If the previous term appears in a denominator, try reciprocals. For a fractional-linear recurrence, fixed points can suggest a ratio transformation. For integer recurrences, parity or residues modulo a small integer can reveal a late term without computing hundreds of steps.
Do not declare periodicity after seeing a few repeated numbers unless you know what state determines the next term. In a second-order recurrence, repeating one term is not enough; you may need a repeated adjacent pair.
Series questions are often recognition questions
TMUA series problems can hide telescoping inside a recurrence, geometric behaviour inside a transformed variable, or a coefficient system inside a binomial expansion. Write a few symbolic terms before reaching for a formula.
For an infinite geometric series, the convergence condition matters as much as the sum formula. A formally correct expression for the sum is irrelevant if the common ratio does not satisfy the required magnitude condition.
FREE PRACTICE
Try these TMUA-style questions
These questions now draw from the current reviewed CF Maths banks, including image and diagram items where appropriate. Open the solution only after you have committed to a route.
QUESTION 1 · Geometric sequences & seriesShow answer & route
Answer: B
Correct answer: \(B\). This item is marked from the validated final 200-question answer key. Revision method: Geometric sequences & series. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: C
Correct answer: \(C\). This item is marked from the validated final 200-question answer key. Revision method: Arithmetic sequences & series. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: B
Correct answer: \(B\). This item is marked from the validated final 200-question answer key. Revision method: Recurrence relations. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: C
Correct answer: \(C\). This item is marked from the validated final 200-question answer key. Revision method: Sequence and series structure. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: B
Correct answer: \(B\). This item is marked from the validated final 200-question answer key. Revision method: Geometric sequences & series. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: C
Correct answer: \(C\). This item is marked from the validated final 200-question answer key. Revision method: Arithmetic sequences & series. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: B
Correct answer: \(B\). This item is marked from the validated final 200-question answer key. Revision method: Recurrence relations. Rework the displayed question carefully and compare your algebra with the keyed option.
Show answer & route
Answer: D
Correct answer: \(D\). This item is marked from the validated final 200-question answer key. Revision method: Arithmetic sequences & series. Rework the displayed question carefully and compare your algebra with the keyed option.
