Sequences & Series

Concept overview

An arithmetic progression (AP) has constant difference d: aₙ = a₁ + (n − 1)d, and sum S_n = (n/2)(2a₁ + (n − 1)d) = (n/2)(a₁ + aₙ). A geometric progression (GP) has constant ratio r: aₙ = a₁ r^(n−1), and sum S_n = a₁(rⁿ − 1)/(r − 1) for r ≠ 1. For |r| < 1, the infinite sum converges to a₁/(1 − r).

Harmonic progression (HP) is the reciprocal of an AP: 1/a₁, 1/a₂, ... form an AP iff a₁, a₂, ... form an HP. The harmonic mean H = 2ab/(a + b) for two numbers.

Three classic means for positive numbers: arithmetic AM = (a + b)/2, geometric GM = √(ab), harmonic HM = 2ab/(a + b). They satisfy AM ≥ GM ≥ HM with equality iff all numbers are equal. This is the AM-GM inequality, and it underlies many optimization problems.

Sum of first n natural numbers is n(n+1)/2; sum of squares is n(n+1)(2n+1)/6; sum of cubes is [n(n+1)/2]². Telescoping sums — where consecutive terms cancel — are a favourite JEE technique for closed-form summation.

Key formulas & concepts

• AP n-th term: aₙ = a₁ + (n − 1)d
• AP sum: S_n = (n/2)(2a₁ + (n − 1)d)
• GP n-th term: aₙ = a₁ r^(n−1)
• GP finite sum: S_n = a₁(rⁿ − 1) / (r − 1)
• GP infinite sum: S∞ = a₁/(1 − r) for |r| < 1
• AM ≥ GM ≥ HM: for positive numbers, equality iff all equal
• Sum of squares: Σₖ₌₁ⁿ k² = n(n+1)(2n+1)/6
• Sum of cubes: Σₖ₌₁ⁿ k³ = [n(n+1)/2]²

Common mistakes to avoid

• Applying infinite GP sum formula with |r| ≥ 1 — the series diverges.
• Confusing AM ≥ GM with GM ≥ AM — always AM ≥ GM for positive numbers.
• Using AP sum when common difference isn't constant.
• Ignoring the equality condition in AM-GM — it holds iff all numbers are equal.

JEE Main relevance

JEE Main asks 1–2 sequences/series problems every paper — usually one AP/GP sum and one AM-GM inequality application.

JEE Advanced relevance

Advanced problems combine multiple progressions (find x if a, b, c are in AP and 1/a, 1/b, 1/c in GP) or use telescoping techniques to derive closed forms for non-trivial sums.

Previous-year questions (PYQ)

Filter the PYQ bank by 'Sequences and Series'. AP-GP mixture problems are the most repeated single archetype.