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Mathematics: Class 10 — Chapter 5: Arithmetic Progressions

5.1 Introduction to Arithmetic Progressions

Definition of an AP

  • An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
  • Each number in the list is called a term.
  • The fixed number added to each term is called the common difference (denoted by d). It can be positive, negative, or zero.

General Form of an AP

  • The general form of an AP is: a, a + d, a + 2d, a + 3d, ... where a is the first term and d is the common difference.
  • Finite AP: An AP containing a finite number of terms, which always has a last term.
  • Infinite AP: An AP that does not have a finite number of terms and has no last term.
5.2 The n-th Term of an AP

Formula for the n-th Term

  • The n-th term (or general term) an of an AP with first term a and common difference d is given by:
    • an = a + (n - 1)d
  • If there are m terms in the AP, then am represents the last term, sometimes denoted by l.
5.3 Sum of First n Terms of an AP

Formulas for Sum (Sn)

  • The sum of the first n terms of an AP is given by:
    • Sn = [n / 2] * [2a + (n - 1)d]
  • If the first term a and the last term l are known, the sum can also be written as:
    • Sn = [n / 2] * (a + l)
  • The sum of the first n positive integers is: Sn = [n(n + 1)] / 2.

Arithmetic Mean

  • If three numbers a, b, and c are in AP, then b is called the arithmetic mean of a and c:
    • b = (a + c) / 2