Arithmetic Sequence Important Points

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, …

5, 10, 15, 20, …

1, 4, 9, 16, 25, …

2,4,8,16, …

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A sequence got by starting with any number and adding a fixed

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

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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

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Consider  4,9,14,19,24,29,…

Consider the consecutive terms 4,9,14

Consider the consecutive terms 14,19,24

Consider the consecutive terms 19,24,29

If a,b,c are three consecutive terms of an AP, then

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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

Xm-Xn = (m-n)d

The difference between any two terms of an arithmetic sequence

is a multiple of common difference.

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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 

Sum

Sn denotes the sum of consecutive n terms of an arithmetic sequence

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