Arithmetic Progressions — Part 1: What an AP is and its nth term: visual lesson, CBSE Class 10 Mathematics (Standard 041 / Basic 241)

Lesson

Arithmetic Progressions, Part 1: Patterns that grow evenly

An auto-rickshaw in Pune charges ₹ $30$ for the first kilometre and ₹ $12$ for each further kilometre: $30, 42, 54, 66, \ldots$

In this lesson you will

Decide whether a list of numbers is an AP and find a and d
Write the general form of an AP
Use aₙ = a + (n − 1)d to find any term
Find which term a number is, or show it is not a term
Find the term from the end and an AP from two of its terms

$a_n = 12n + 18$

Each extra kilometre adds the same ₹$12$: the fares form an arithmetic progression.

Concept

What you will be able to do

Decide whether a list is an arithmetic progression (AP)
Find the first term $a$ and common difference $d$
Use $a_n = a + (n - 1)d$ for any term
Decide which term a number is, or whether it is a term at all
Find terms from the end and an AP from two of its terms

Diagram

What is an arithmetic progression?

An AP has one fixed difference, which may be negative or zero.
List	Differences	AP?
$5, 9, 13, 17, \ldots$	$4, 4, 4$	yes, $d = 4$
$20, 14, 8, 2, \ldots$	$-6, -6, -6$	yes, $d = -6$
$7, 7, 7, 7, \ldots$	$0, 0, 0$	yes, $d = 0$
$1, 4, 9, 16, \ldots$	$3, 5, 7$	no

An arithmetic progression is a list of numbers in which each term after the first is obtained by adding a fixed number $d$ , the common difference: $a_{n+1} - a_n = d$ for every $n$ .

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