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SingaporeMaths

Sequences and Series

8 dot points across 8 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do arithmetic progressions grow, and how do we sum them?

Use the formulae for the nth term and the sum of the first n terms of an arithmetic progression, and solve problems involving arithmetic sequences and series
A focused answer to the H2 Mathematics outcome on arithmetic progressions. The nth term and sum formulae, finding the first term and common difference from given conditions, and applying APs to worded problems.
8 min answer →

How do we expand a binomial raised to a rational power, and when is the expansion valid?

Expand (1 + x) to the power n for rational n as a series, state the range of validity, and use the expansion to obtain approximations
A focused answer to the H2 Mathematics outcome on the binomial expansion for rational index. The general series, the range of validity, handling expressions not in standard form, and using the expansion for approximations.
9 min answer →

What does it mean for a series to converge, and how do we reason about the behaviour of a sequence as n grows?

Describe the behaviour of a sequence as n tends to infinity, determine the convergence of a geometric series, and interpret the limit of a sequence or partial sum
A focused answer to the H2 Mathematics outcome on convergence. The behaviour of a sequence as n tends to infinity, the convergence condition for a geometric series, and interpreting limits of sequences and partial sums.
9 min answer →

How do geometric progressions grow, and when does their sum converge?

Use the formulae for the nth term and the sum of a geometric progression, determine convergence, and find the sum to infinity of a convergent geometric series
A focused answer to the H2 Mathematics outcome on geometric progressions. The nth term and sum formulae, the condition for convergence, the sum to infinity, and applications to growth and decay problems.
9 min answer →

How does proof by mathematical induction establish a result for every positive integer?

Use the principle of mathematical induction to prove statements about sums of series, divisibility and other results indexed by the positive integers
A focused answer to the H2 Mathematics outcome on mathematical induction. The base case, the inductive step, writing a rigorous proof, and applying induction to series sums and divisibility results.
9 min answer →

How does a telescoping sum collapse, and how do we exploit it to evaluate a series?

Use the method of differences, including the use of partial fractions, to find the sum of a series whose terms telescope, and deduce the sum to infinity where it exists
A focused answer to the H2 Mathematics outcome on the method of differences. Writing a term as a difference (often via partial fractions), cancelling the telescoping sum, and deducing the sum to infinity.
10 min answer →

How do recurrence relations define a sequence, and how do we find or verify a closed form?

Use recurrence relations to generate sequences, find and verify a conjectured formula for the nth term, and analyse long-term behaviour
A focused answer to the H2 Mathematics outcome on recurrence relations. Generating terms from a recurrence, conjecturing and verifying a closed form, finding a limiting value, and recognising arithmetic and geometric recurrences.
9 min answer →

How does sigma notation express a sum, and which standard results let us evaluate it?

Use sigma notation and the standard results for the sums of integers, squares and cubes, and the linearity of summation, to evaluate finite series
A focused answer to the H2 Mathematics outcome on sigma notation. Reading and writing sums in sigma notation, the standard results for sums of integers, squares and cubes, linearity, and adjusting limits.
9 min answer →