Arithmetic Progression — Part 1: Common difference and the general term: visual lesson, ICSE Class 10 Mathematics

Lesson

Arithmetic Progression, Part 1

An arithmetic progression (AP) grows or shrinks by the same amount at every step. Plotted against the term number, its terms lie on a straight line.

In this lesson you will

Decide whether a sequence is an arithmetic progression and find its common difference
Write the general (nth) term of an AP and find any term
Find which term a given number is, and the number of terms of a finite AP
Find an AP from two conditions, and work with three terms in AP

$T_n = 5n - 2$

Each step adds the same $5$: the terms of an AP lie on a straight line.

Concept

What you will be able to do

Decide whether a sequence is an AP and find its common difference $d$
Write the general term $T_n$ and find any term
Find which term a number is, and how many terms an AP has
Find an AP from two conditions; handle three terms in AP

Concept

What is an arithmetic progression?

A sequence is a list of numbers in order: $T_1, T_2, T_3, \ldots$ It is an AP if every term after the first is the term before it plus a fixed number $d$ , the common difference.

T_{n+1} - T_n = d \text{ for every } n

Same difference every time: an AP. Changing differences: not an AP.
Sequence	Differences $T_2 - T_1$ , $T_3 - T_2$ , $T_4 - T_3$	AP?
$4, 11, 18, 25$	$7, 7, 7$	yes, $d = 7$
$30, 24, 18, 12$	$-6, -6, -6$	yes, $d = -6$
$5, 5, 5, 5$	$0, 0, 0$	yes, $d = 0$
$2, 3, 5, 8$	$1, 2, 3$	no: the differences change

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