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# 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, …

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

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

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

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.

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