Arithmetic Progression — Study Notes

Overview

Arithmetic Progression (AP) is one of the most predictable and high-scoring topics in the KAR TET Paper II Mathematics section. It forms the foundation for understanding sequences and series, and questions typically test your ability to identify patterns, find specific terms, and calculate sums.

This topic connects directly to real-life situations—calculating monthly savings, arranging seats in auditoriums, or understanding patterns in nature. From an exam perspective, expect 2–3 direct questions that reward formula memorisation and careful substitution. Mastering AP also builds your confidence for related concepts like geometric progression in higher studies.

The key to success here is understanding what makes a sequence "arithmetic" and knowing when to apply which formula. Most questions become straightforward once you identify the first term (a) and common difference (d).

Key Concepts

• **Arithmetic Progression defined**: A sequence where each term after the first is obtained by adding a fixed number (common difference) to the previous term. Example: 3, 7, 11, 15, ... has common difference 4.

• **Common difference (d)**: The constant value added to get the next term. Calculate as d = (any term) − (previous term). If d > 0, AP is increasing; if d < 0, AP is decreasing; if d = 0, all terms are equal.

• **First term (a)**: The starting point of the AP. Every formula revolves around knowing 'a' and 'd'.

• **General (nth) term**: Any term's position can be found without listing all previous terms. This is the core skill examiners test.

• **Finite vs Infinite AP**: Finite AP has a definite last term (like 2, 5, 8, 11); infinite AP continues indefinitely (like 1, 2, 3, 4, ...).

• **Three consecutive terms**: If three numbers are in AP, the middle term equals the average of the other two. So if a, b, c are in AP, then b = (a + c)/2, or equivalently, 2b = a + c.

• **Sum of AP**: The total of all terms from the first to the nth term. This is frequently tested using both forms of the sum formula.

Formulas / Key Facts

**nth Term of AP** aₙ = a + (n − 1)d

• aₙ = nth term, a = first term, n = position, d = common difference
• Use when: finding any specific term or checking if a number belongs to the AP

**Sum of First n Terms (Form 1)** Sₙ = n/2 × [2a + (n − 1)d]

• Use when: you know a, d, and n

**Sum of First n Terms (Form 2)** Sₙ = n/2 × (a + l)