Chapter 5 — Arithmetic Progressions

An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number called the common difference (d) to the preceding term; the nth term is given by an = a + (n − 1)d and the sum of the first n terms by Sn = n/2 [2a + (n − 1)d].

NCERT Class 10 Maths Chapter 5 covers Arithmetic Progressions — sequences where each term differs from the previous by a constant value called the common difference (d). The chapter introduces the general form a, a+d, a+2d, …, derives the formula for the nth term (an = a + (n−1)d), and develops two sum formulas: Sn = n/2 [2a + (n−1)d] and Sn = n/2 (a + l), where l is the last term. Real-life examples include salary increments, savings schemes, and flower-bed rows, grounding abstract concepts in everyday contexts.

1. 01An AP is defined as a list of numbers where each term is obtained by adding a fixed common difference d to the preceding term; d can be positive, negative, or zero.
2. 02The 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.
3. 03The nth term (general term) of an AP is given by an = a + (n−1)d; the last term of a finite AP is sometimes denoted l.
4. 04The sum of the first n terms is Sn = n/2 [2a + (n−1)d], or equivalently Sn = n/2 (a + l) when the last term l is known.
5. 05A list is an AP if and only if the difference between every pair of successive terms is the same, i.e., ak+1 − ak is constant for all k.
6. 06The nth term can also be found as the difference of consecutive partial sums: an = Sn − Sn−1.