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what is arithmetic progression

Arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is always the same.

What is Arithmetic Progression?

An arithmetic progression is a list like 2,5,8,11,…2,5,8,11,\dots 2,5,8,11,… where each new term is obtained by adding a fixed number (called the common difference) to the previous term.

If the first term is aaa and the common difference is ddd, a general AP looks like:

a, a+d, a+2d, a+3d, …a,;a+d,;a+2d,;a+3d,;\dots a,a+d,a+2d,a+3d,…

Key Terms (Quick Scoop)

  • First term: aaa.
  • Common difference: ddd, the fixed amount you add or subtract each time.
  • nnnth term: The formula for the nnnth term is

an=a+(n−1)da_n=a+(n-1)dan​=a+(n−1)d

A simple example: in 3,7,11,15,…3,7,11,15,\dots 3,7,11,15,…, we have a=3a=3a=3 and d=4d=4d=4.

Common Formulas You’ll See

  • nnnth term:

an=a+(n−1)da_n=a+(n-1)dan​=a+(n−1)d

  • Sum of first nnn terms:

Sn=n2 [2a+(n−1)d]S_n=\frac{n}{2},[2a+(n-1)d]Sn​=2n​[2a+(n−1)d]

These formulas are heavily used in school exams and competitive tests when working with sequences.

Tiny Real-Life Picture

You see arithmetic progressions when something increases (or decreases) by a fixed amount regularly:

  • A salary that goes up by a fixed amount each year.
  • Even numbers 2,4,6,8,…2,4,6,8,\dots 2,4,6,8,… with d=2d=2d=2.
  • Natural numbers 1,2,3,4,…1,2,3,4,\dots 1,2,3,4,… with d=1d=1d=1.

Quick Reference Table (Core Ideas)

html

<table>
  <tr>
    <th>Concept</th>
    <th>Arithmetic Progression (AP)</th>
  </tr>
  <tr>
    <td>Definition</td>
    <td>A sequence where the difference between consecutive terms is constant.[web:1][web:7][web:5]</td>
  </tr>
  <tr>
    <td>General form</td>
    <td>a, a + d, a + 2d, a + 3d, …[web:1][web:5]</td>
  </tr>
  <tr>
    <td>Common difference</td>
    <td>d (fixed number added each step).[web:1][web:7][web:9]</td>
  </tr>
  <tr>
    <td>n-th term</td>
    <td>a<sub>n</sub> = a + (n − 1)d.[web:1][web:5][web:9]</td>
  </tr>
  <tr>
    <td>Sum of first n terms</td>
    <td>S<sub>n</sub> = n/2 · [2a + (n − 1)d].[web:1][web:5][web:8]</td>
  </tr>
</table>

TL;DR: Arithmetic progression is a number sequence with a constant step between terms, like 5,8,11,14,…5,8,11,14,\dots 5,8,11,14,…, defined by first term aaa, difference ddd, and formulas for the nnnth term and sum.

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