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Sequences and Series

Two conditions on an AP or a GP, then a term or a sum; the series pages ask you to find the kth term first. A large share has numeric answers.

Questions in the bank
221
q/paper in 2025–26
1.65
Numeric answer
37%
Notes pages
8

Tier: Cornerstone

When you’ll see it

Terms said to be in AP, GP or HP, a sum to n terms or to infinity, or a series whose kth term must be found.

How this chapter is tested

The chapter runs from progressions to series. AP and GP questions usually give two conditions and ask for a later term or a sum; the work is two equations in a and d, or in a and r, solved by dividing one by the other.

The series pages cost more time. Σk, Σk² and Σk³ handle a polynomial kth term; telescoping handles a term that splits as f(k) − f(k + 1); multiplying by the ratio and subtracting handles an arithmetico-geometric series. In each, finding the kth term is the real step.

Common terms of two progressions are often set as numeric-answer questions. GPs also hide inside functions such as f(x + y) = f(x)f(y) and inside recurrences. AM–GM gives the least and greatest values here and returns in quadratic equations, the binomial theorem and calculus.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Arithmetic progressions and common terms

    aₙ = a + (n − 1)d, Sₙ = n/2 (2a + (n − 1)d); the common terms of two APs with whole-number steps form an AP with step lcm(d₁, d₂), if they share any term at all.

  • Geometric progressions

    aₙ = arⁿ⁻¹; divide two conditions to find r; spot a GP in a function rule or a recurrence.

  • Means and AM–GM

    2b = a + c in AP, b² = ac in GP, b = 2ac/(a + c) in HP; A ≥ G ≥ H for positive numbers; AM–GM for extremes.

  • Infinite GPs

    Sum a/(1 − r) for |r| < 1; the squares form a GP with ratio r².

  • Sums by standard formulas

    Σk = n(n + 1)/2, Σk² = n(n + 1)(2n + 1)/6, Σk³ = (n(n + 1)/2)²; find Tₖ from differences.

  • Telescoping sums

    Write Tₖ as f(k) − f(k + 1) so the middle cancels, using partial fractions, surds or factorials.

  • Arithmetico-geometric and exponential series

    Multiply by r and subtract; Σ 1/k! gives e, and Σ xᵏ/k from k = 1 gives −ln(1 − x).

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Steps, not terms

    From the pth term to the qth there are q − p steps. With n means inserted between two numbers there are n + 1 steps, not n.

  • The rejected ratio

    A quadratic in r often has roots t and 1/t. An increasing GP of positive terms keeps the root above 1, and a sum to infinity rejects any root with |r| ≥ 1.

  • The ½ in a telescoping split

    1/(k(k + 2)) = ½(1/k − 1/(k + 2)). Dropping the ½ doubles the answer, and a gap of two leaves two terms at each end, not one.

  • AM–GM without the equality case

    The bound is the answer only if the equal-pieces point is allowed: positive, and inside any stated range.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Sequences and Series notes

Drill every Sequences and Series question

221 questions from the bank, across 8 subtopics.

Drill the 221 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

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