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Schooling · Class 10 · Mathematics · Arithmetic Progressions · Chapter on a page

CBSE Class 10 · Mathematics · NCERT Mathematics

Arithmetic Progressions: 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 26 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.

This note's current text has no applied fact-check on record.

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 Arithmetic Progressions: Key ideas, Formulas and facts, Worked examples, Watch out forKey ideasAn arithmetic progression is a list…The first term of an AP is usually…The nth term (also called the…If d > 0 the AP is increasing; if d…The sum of the first n terms…To check whether a number belongs to…Three numbers are in AP if and only…Formulas and factsnth termSum of first n terms (form 1)Sum of first n terms (form 2)Relation between Sₙ and aₙCondition for three numbers to be in…Worked examplesFinding the 20th termSum of first 15 termsWhich term equals a given value?Watch out forUsing n instead of (n − 1) in the…Forgetting the factor of 1/2 in the…Taking d as positive when the…Assuming every sequence with a…Mixing up aₙ and Sₙ in problemsArithmeticProgressions
The mind map as a list
  • Key ideas
    • An arithmetic progression is a list…
    • The first term of an AP is usually…
    • The nth term (also called the…
    • If d > 0 the AP is increasing; if d…
    • The sum of the first n terms…
    • To check whether a number belongs to…
    • Three numbers are in AP if and only…
  • Formulas and facts
    • nth term
    • Sum of first n terms (form 1)
    • Sum of first n terms (form 2)
    • Relation between Sₙ and aₙ
    • Condition for three numbers to be in…
  • Worked examples
    • Finding the 20th term
    • Sum of first 15 terms
    • Which term equals a given value?
  • Watch out for
    • Using n instead of (n − 1) in the…
    • Forgetting the factor of 1/2 in the…
    • Taking d as positive when the…
    • Assuming every sequence with a…
    • Mixing up aₙ and Sₙ in problems

What this chapter is about

An arithmetic progression (AP) is a sequence of numbers where each term after the first is obtained by adding a fixed number to the previous term. This fixed number is called the common difference. Examples surround us: monthly savings that increase by a fixed amount, rows of seats in a stadium, or stacks of objects with a regular pattern.

Key ideas

  • An arithmetic progression is a list of numbers where the difference between any two consecutive terms is constant; this constant is the common difference, denoted by d.
  • The first term of an AP is usually denoted by a; the sequence then runs a, a + d, a + 2d, a + 3d, and so on.
  • The nth term (also called the general term) of an AP is given by aₙ = a + (n − 1)d; this lets you jump directly to any term.
  • If d > 0 the AP is increasing; if d < 0 it is decreasing; if d = 0 every term equals a.
  • The sum of the first n terms, denoted Sₙ, can be found using formulas involving a, d and n.
  • To check whether a number belongs to an AP, set aₙ equal to that number and solve for n; if n is a positive integer, the number is a term of the AP.
  • Three numbers are in AP if and only if twice the middle number equals the sum of the other two.

Formulas and facts to remember

  • nth term: aₙ = a + (n − 1)dMeaning: start at a and add d exactly (n − 1) times.
  • Sum of first n terms (form 1): Sₙ = n/2 × [2a + (n − 1)d]Meaning: multiply the number of terms by the average of the first and last terms (expressed using a and d).
  • Sum of first n terms (form 2): Sₙ = n/2 × (a + l), where l is the last termMeaning: when the last term is known, half of n times (first + last) gives the sum.
  • Relation between Sₙ and aₙ: aₙ = Sₙ − Sₙ₋₁ for n ≥ 2Meaning: the nth term is the difference between the sum up to n terms and the sum up to (n − 1) terms.
  • Condition for three numbers to be in AP: If x, y, z are in AP then 2y = x + z.

Common mistakes

  • Using n instead of (n − 1) in the nth-term formula → Remember you add d only (n − 1) times after the first term.
  • Forgetting the factor of 1/2 in the sum formula → Write the formula first, then substitute values carefully.
  • Taking d as positive when the sequence is decreasing → Compute d = (second term) − (first term); keep the sign.
  • Assuming every sequence with a pattern is an AP → Check that the difference between consecutive terms is constant throughout.
  • Mixing up aₙ and Sₙ in problems → Read whether the question asks for a particular term or the sum of terms.

Quick revision

  • AP: a, a + d, a + 2d, … where d is the common difference.
  • nth term: aₙ = a + (n − 1)d.
  • Sum of n terms: Sₙ = n/2 × [2a + (n − 1)d] or n/2 × (a + l).
  • To check membership, solve aₙ = given value for n; n must be a positive integer.
  • d can be positive, negative or zero.