Definition of AP

Learn the definition of an Arithmetic Progression (AP), understand the common difference, identify APs, and practice with simple, student-friendly examples.

1. Introduction

An Arithmetic Progression (AP) is a sequence where each term is obtained by adding the same fixed number to the previous term. This fixed number is called the common difference.

In simple words, an AP grows or reduces at a constant rate.

2. What Is an Arithmetic Progression?

A sequence \(a_1, a_2, a_3, ...\) is an AP if:

\(a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = d\)

where d is the common difference.

Examples:

  • 2, 5, 8, 11, ... (d = 3)
  • 10, 7, 4, 1, ... (d = -3)
  • 4, 4, 4, 4, ... (d = 0)

3. Understanding the Common Difference (d)

The common difference tells us how the AP is changing.

Formula:

\(d = a_{n} - a_{n-1}\)

3.1. Types of Common Difference

  • Positive d: AP increases (example: 5, 9, 13, ...)
  • Negative d: AP decreases (example: 20, 15, 10, ...)
  • Zero d: All terms are equal (example: 7, 7, 7, ...)

3.2. Examples (Finding d)

Sequence: 12, 15, 18, 21

\(d = 15 - 12 = 3\)

Sequence: 30, 25, 20, 15

\(d = 25 - 30 = -5\)

4. How to Check if a Sequence Is an AP

To verify whether a sequence is an AP:

  1. Find the differences between consecutive terms.
  2. If all the differences are equal, the sequence is an AP.
  3. If not equal, then it is not an AP.

4.1. Examples

Check if 3, 7, 11, 15 is an AP:

Differences: 7−3 = 4, 11−7 = 4, 15−11 = 4

All differences equal → AP with \(d = 4\)

Check if 2, 5, 9, 14 is an AP:

Differences: 3, 4, 5 → Not equal → Not an AP

5. Real-Life Meaning of AP

APs appear often in real-life situations:

  • Daily salary increase
  • Climbing stairs (same height each step)
  • Plant growth increasing by a fixed amount
  • Saving money regularly each month
  • Seat numbers increasing row by row

6. Common Mistakes

  • Assuming a sequence is AP by looking at only the first two terms.
  • Incorrect subtraction while finding common difference.
  • Misunderstanding that AP must always increase—it can decrease too.
  • Confusing AP with other patterns like doubling (which is GP).

7. Quick Practice

Identify whether the following sequences are APs. If yes, find the common difference:

  1. 4, 10, 16, 22
  2. 100, 90, 80, 70
  3. 2, 6, 18, 54
  4. 7, 7, 7, 7

8. Summary

  • An AP is a sequence with a constant difference between terms.
  • The common difference may be positive, negative, or zero.
  • To check an AP, compare consecutive term differences.
  • APs appear naturally in many daily patterns.