Chapter 5 — Arithmetic Progressions

Chapter 5 of the Class 10 Maths NCERT textbook, "Arithmetic Progressions", covers lists of numbers where each term is obtained by adding a fixed common difference (d) to the one before, with the nth term given by an = a + (n − 1)d and the sum of n terms by Sn = n/2 [2a + (n − 1)d].

• The idea of a common difference — An arithmetic progression grows by adding the same fixed amount, the common difference, at every step. This difference can be positive, negative, or zero, and stays constant between all successive terms.
• Reaching any term directly — Rather than listing every term, the nth-term formula lets you jump straight to a term's value from the first term and the common difference — handy for finding the 100th term without writing the others.
• Summing the terms — The chapter derives a formula to add the first n terms at once, with an equivalent version using the last term. This turns long additions into a single quick calculation.

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.