Examples and practice
Sequences and series
Back to lessonPractice section 1 · Questions 1–6
Generate terms and describe a formula · Separate monotonic behaviour from convergence · +4 more
Practice section 2 · Questions 7–12
Read an arithmetic progression and form its nth term · Find two specified arithmetic terms · +4 more
Practice section 3 · Questions 13–18
Recognise a common ratio · Form the nth term of a geometric progression · +4 more
Practice section 4 · Questions 19–24
Check a GP proof and a constant sequence · Recover terms and check the exceptional first term · +4 more
Practice section 5 · Questions 25–30
Evaluate three finite sums · Find the first harmonic partial sum above four · +4 more
Practice section 6 · Questions 31–36
Sum twenty arithmetic terms · Recover an AP from two consecutive blocks of sums · +4 more
Practice section 7 · Questions 37–42
Total the doubling reward on a chessboard · Evaluate a finite geometric sum above ratio one · +4 more
Practice section 8 · Questions 43–48
Resolve the constant cases left out of the table · Distinguish term limits from series limits · +4 more
Practice section 9 · Questions 49–54
Split an infinite GP into equal prefix and tail sums · Combine polynomial and geometric finite sums · +4 more
- Split an infinite GP into equal prefix and tail sums
- Combine polynomial and geometric finite sums
- Unfold a geese-population recurrence in two directions
- Fit an exponential-linear recurrence and sum its terms
- Relate an exponential recurrence to its two fixed points
- Use an AP of sector areas and a GP condition
Practice section 10 · Questions 55–60
Compare AP and GP travel totals in a chase · Count travelled legs in a bouncing-ball model · +4 more
Practice section 11 · Questions 61–66
Exclude a digit using arithmetic sums · Sum increasing interest-free instalments · +4 more
Practice section 12 · Questions 67–72
Split mixed sigma sums into arithmetic and geometric parts · Change a summation range with two partial sums · +4 more
Practice section 13 · Questions 73–78
Use a logarithmic fixed-point graph to compare terms · Compare geometric and arithmetic instrument bars · +4 more
- Use a logarithmic fixed-point graph to compare terms
- Compare geometric and arithmetic instrument bars
- Combine a capped geometric plan with arithmetic increments
- Track beginning-of-month deposits and monthly interest
- Evaluate a sum with three different structures
- Recover the square-sum formula from a cubic identity
Practice section 14 · Questions 79–84
Match the final odd square to its summation index · Subtract the correct prefix from a finite sum · +4 more
- Match the final odd square to its summation index
- Subtract the correct prefix from a finite sum
- Express consecutive pair products using square and linear sums
- Rationalise a radical sum and distinguish terms from partial sums
- Shift both bounds before applying a prefix formula
- Separate odd-index and even-index denominator terms
Practice section 15 · Questions 85–90
Compare positive denominators to prove a strict lower bound · Sum the rows of a triangular labelled board · +4 more
- Compare positive denominators to prove a strict lower bound
- Sum the rows of a triangular labelled board
- Distinguish a sequence of geometric sums from its own sum
- Recognise factorial growth from the recurrence
- Treat fixed points as candidate limits before proving decrease
- Count elapsed updates separately from the week label
Practice section 16 · Questions 91–96
Use an iteration to approximate the specified cubic root · Unfold a loss-then-top-up recurrence in two directions · +4 more
- Use an iteration to approximate the specified cubic root
- Unfold a loss-then-top-up recurrence in two directions
- Relate separated arithmetic terms to a geometric progression
- Recover terms from partial sums and compare a geometric tail
- Combine geometric heights with a weighted class mean
- Compare a geometric training total with an arithmetic total
Practice section 17 · Questions 97–98
Remove grain before growth and retain the exact supply bound · Add a late fee after interest and count the charging cycles