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Schooling · Class 11 · Mathematics · Sequences and Series · Chapter on a page

CBSE Class 11 · Mathematics · NCERT Mathematics

Sequences and Series: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Sequences and Series: Key ideas, Formulas and facts, Watch out forKey ideasA sequence is a function whose…In an arithmetic progression (AP)…In a geometric progression (GP), the…The arithmetic mean (AM) of two…The geometric mean (GM) of two…A series is the sum of terms of a…The sum of an infinite GP with |r| <…Formulas and factsFor an Arithmetic Progression with…nth termSum of first n termsAM = (a + b)/2For a Geometric Progression with…S∞ = a/(1 − r)GM = √(ab)Watch out forUsing the GP sum formula with r = 1Confusing the nth term with the sum…Applying the infinite GP formula…Forgetting that r can be negative in…Calculating GM of negative numbers…Sequences and Series
The mind map as a list
  • Key ideas
    • A sequence is a function whose…
    • In an arithmetic progression (AP)…
    • In a geometric progression (GP), the…
    • The arithmetic mean (AM) of two…
    • The geometric mean (GM) of two…
    • A series is the sum of terms of a…
    • The sum of an infinite GP with |r| <…
  • Formulas and facts
    • For an Arithmetic Progression with…
    • nth term
    • Sum of first n terms
    • AM = (a + b)/2
    • For a Geometric Progression with…
    • S∞ = a/(1 − r)
    • GM = √(ab)
  • Watch out for
    • Using the GP sum formula with r = 1
    • Confusing the nth term with the sum…
    • Applying the infinite GP formula…
    • Forgetting that r can be negative in…
    • Calculating GM of negative numbers…

What this chapter is about

A sequence is an ordered list of numbers following a definite rule, where each number is called a term. When we add the terms of a sequence together, we get a series. This chapter introduces you to the systematic study of patterns in numbers—how they grow, shrink, or change according to fixed rules.

Key ideas

  • A sequence is a function whose domain is the set of natural numbers or a subset {1, 2, 3, …, k} of it (a finite sequence); we denote the terms as a₁, a₂, a₃, ... or simply {aₙ}.
  • In an arithmetic progression (AP), the difference between consecutive terms is constant; this constant is called the common difference, denoted d. So aₙ₊₁ − aₙ = d for all n.
  • In a geometric progression (GP), the ratio of consecutive terms is constant; this constant is called the common ratio, denoted r. So aₙ₊₁ / aₙ = r for all n, where aₙ ≠ 0.
  • The arithmetic mean (AM) of two numbers a and b is (a + b)/2; it lies exactly midway between them on the number line.
  • The geometric mean (GM) of two positive numbers a and b is √(ab); for positive numbers, GM ≤ AM always.
  • A series is the sum of terms of a sequence; when the sequence is finite, the series has a definite sum, and when infinite, convergence must be checked.
  • The sum of an infinite GP with |r| < 1 converges to a/(1 − r), where a is the first term.

Formulas and facts to remember

  • For an Arithmetic Progression with first term a and common difference d:
  • nth term: aₙ = a + (n − 1)d — each term is the first term plus (n − 1) times the common difference.
  • Sum of first n terms: Sₙ = n/2 × [2a + (n − 1)d] or equivalently Sₙ = n/2 × (a + l), where l is the last term.
  • Arithmetic mean of a and b: AM = (a + b)/2.
  • For a Geometric Progression with first term a and common ratio r:
  • nth term: aₙ = a × r^(n − 1) — each term is the first term multiplied by r raised to (n − 1).
  • Sum of first n terms: Sₙ = a(rⁿ − 1)/(r − 1) when r ≠ 1; Sₙ = na when r = 1.
  • Sum of infinite GP (|r| < 1): S∞ = a/(1 − r).
  • Geometric mean of two positive numbers a and b: GM = √(ab).
  • Relation between AM and GM:
  • For any two positive numbers a and b: AM ≥ GM, with equality only when a = b.

Common mistakes

  • Using the GP sum formula with r = 1 → when r = 1, all terms are equal, so Sₙ = na; the standard formula has division by zero.
  • Confusing the nth term with the sum of n terms → aₙ gives one specific term, while Sₙ gives the total of all terms up to that point.
  • Applying the infinite GP formula when |r| ≥ 1 → the series diverges; the formula a/(1 − r) only works when |r| < 1.
  • Forgetting that r can be negative in a GP → a sequence like 3, −6, 12, −24, ... is a GP with r = −2; the signs alternate.
  • Calculating GM of negative numbers directly using √(ab) → the standard GM formula applies to positive numbers; for negatives, interpret carefully or state it is undefined for real numbers.

Quick revision

  • AP: constant difference between terms; nth term = a + (n − 1)d; sum = n/2 × (first + last).
  • GP: constant ratio between terms; nth term = a × r^(n − 1); sum of infinite GP (|r| < 1) = a/(1 − r).
  • AM of a and b is (a + b)/2; GM of positive a and b is √(ab); AM ≥ GM always.
  • Check whether a sequence is AP by verifying aₙ₊₁ − aₙ is constant; check GP by verifying aₙ₊₁/aₙ is constant.
  • To find the number of terms, use the nth term formula and solve for n.