Arithmetic Progressions

1. Chapter at a glance

• An arithmetic progression (AP) is a list of numbers in which each term (except the first) is obtained by adding a fixed number d (the common difference) to the preceding term.
• 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 (d may be positive, negative or zero).
• A finite AP has a last term; an infinite AP does not.
• The nth term (general term) of an AP is given by an = a + (n − 1)d.
• The sum of the first n terms of an AP is given by Sn = n/2 [2a + (n − 1)d] or Sn = n/2 (a + l) where l is the last term.
• A list forms an AP if and only if the difference between any two consecutive terms is constant, i.e., ak+1 − ak = d for all k.
• To find d it is sufficient to subtract any one term from its succeeding term.

2. Definitions, theorems and results

• Definition (NCERT): An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term. This fixed number is called the common difference of the AP.
• Result: In an AP with first term a and common difference d, the nth term is an = a + (n − 1)d. (an is also called the general term.)
• Result: If an AP has m terms, then am represents the last term, sometimes denoted by l.
• Result: The sum of the first n terms of an AP is Sn = n/2 [2a + (n − 1)d].
• Result: If l is the last term of a finite AP of n terms, then Sn = n/2 (a + l).
• Result (derived): an = Sn − Sn−1.
• No theorems/axioms/ lemmas are stated in the chapter that require a formal proof in the exam; the formulas are obtained by direct derivation or by the method illustrated with the sum 1 + 2 + … + n.

3. Formula sheet

4. Solved-example patterns

• Check whether a given list is an AP and find d (if it is): Compute successive differences ak+1 − ak and verify they are equal. If equal, d equals that constant value. Write next terms by adding d repeatedly.
• Find the nth term given a and d: Substitute directly into an = a + (n − 1)d.
• Find which term equals a given number (or whether a number belongs to the AP): Set a + (n − 1)d equal to the given number and solve for n; n must be a positive integer.
• Find a and d when two terms are given: Write the two equations a + (p − 1)d = given value for pth term and a + (q − 1)d = given value for qth term; solve the linear pair.
• Find the term from the end: First obtain total number of terms n from an = l, then compute the required term as a(n − k + 1) where k is the position from the end (or reverse the AP).
• Find number of terms satisfying a condition (e.g., two-digit multiples of 3): Express the sequence as an AP, set an equal to the last qualifying term and solve for n.
• Find Sn given a, d and n (or given three of the four quantities): Substitute into Sn = n/2 [2a + (n − 1)d] or Sn = n/2 (a + l). When Sn and two other quantities are known, solve for the unknown.
• Find a or d or n when Sn and other data are given: Use the appropriate Sn formula and solve the resulting equation (may yield quadratic; check both roots for validity).

5. Common mistakes and exam pitfalls

• Subtracting terms in the wrong order when finding d (must compute later term minus earlier term).
• Forgetting that n must be a positive integer; discarding non-integer or negative solutions.
• Sign errors when d is negative (especially when finding terms from the end or solving for n).
• Using Sn = n/2 (a + l) when l is not known or when the AP is infinite.
• Assuming every sequence with a pattern is an AP (e.g., squares, geometric sequences fail the constant-difference test).
• Missing that two values of n may both be valid when the quadratic equation appears (sum of terms after a certain point may be zero).
• Confusing the 11th term from the end with the (n − 11)th term instead of the (n − 10)th term.