The concept of arithmetic sequence also called

**arithmetic progression**or**AP**, it's a common difference, properties, general term, number of terms,the sum to n term is discussed.**High School Study Material**

Class X (Class 10) Mathematics

Chapter 1

Arithmetic Sequence

A set of numbers written as First second third and so on according to a particular rule is called a number sequence.

Example:

2, 4, 6, 8,10, …

Example:

2, 4, 6, 8,10, …

5, 10, 15, 20, …

1, 4, 9, 16, 25, …

2,4,8,16, …

number repeatedly is called Arithmetic Sequence or

**AP**.
The fixed number is called common difference and it is denoted by ‘d’

4 ,7 ,10 ,13,…

15, 13,11, 9,…

In the first arithmetic sequence, d = 3

In the second arithmetic sequence , d = -2

Consider the arithmetic sequence (AP)

6,10,14,18,…

15, 13,11, 9,…

First term , X1 = 6

Second term, X2 = 10

Fourth term, X4 = 18

d = X2 - X1 = 10-6 = 4

= X3 - X2 = 14-10 = 4

= X4 - X3 = 18-14 = 4

The difference between any two consecutive terms is the common difference

In an

**arithmetic sequence**

X2 = X1 + d

X3 = X1 + 2d

X4 = X1 + 3d

X5 = X1 + 4d

X10 = X1 + 9d

X14 = X1 + 13d

X5 = X3 + 2d

X10 = X4+ 6d

X25 = X9 + 16d

X3 = X10 – 7d

X5 = X15 – 10d

X7 = X20 -13d

X10-X5 = (10-5)d= 5d

X15-X7 =(15-7)d = 8d

X20-X8 = (20-8)d= 12d

The difference between any two terms of an

**arithmetic sequence**
is a multiple of common difference.

Consider the arithmetic sequence1,4,7,10,13,….

1+4+7 = 12 = 3×4= 3×middle term

1+4+7+10+13= 35 = 5×7 = 5×middle term

The sum of n terms of an arithmetic sequence = n × middle term

Where n is an odd number.

General term AP

Xn is the nth term or general term or algebraic form of

an arithmetic sequence

Xn = f – d + nd

Where f is the first term

The general term of an arithmetic sequence is of the form an+b.

The coefficient of n is the common difference

Eg.

If the nth term of an arithmetic sequence is 3n+2, its common difference is 3.

The number of terms or position of a term is given by

If we divide any term of an arithmetic sequence with common difference, we get the same remainder

If we divide any term of an arithmetic sequence with common difference, we get the same remainder

Sum

Where n is the number of terms, Xn is the last term and f is the first term.

Sum of first n natural numbers

Sum of first n even natural numbers = n(n+1)

Sum of first n odd natural numbers = n2

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