Arithmetic Progressions Class 10 Notes Maths Chapter 5

CHAPTER AT A GLANCE

1. Arithmetic Progression (AP)
Consider
(i) 1,2,3,4, ……….
(ii) 3, 3, 3, 3, ……….
(i) and (ii) are the sequence of numbers, each number in these sequences is called a term.

An arithmetic progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number ‘d’ to the preceeding term, except the first term.
The fixed number is caled the common difference. It can be positive, negative or zero.

Any Arithmetic progression can be represented as :
a,a+d,a+2d,a+3d, ……
where ‘a’ is the first term & ‘d’ is the common difference.
Arithmetic progressions which does not have a last term are called Infinite Arithmetic Progression. e.g.;
6, 9, 12, 15, ……………..

2. Formula for Common Difference (d)

A sequence of numbers a, a1,a2,a3… is an AP if the difference a2-a1,a3-a2,a4-a3 ….
gives the same value, i.e. if(ak+1 -ak) is the same for different values of k. The difference (ak+l -ak) is called common difference denoted by d.
Here ak + 1 & ak are the (k + 1)th & kth terms respectively.
:. d =a2 -a1 =a3-a2 = a4 -a3……..

3. nth Term (or General Term) of an Arithmetic Progressions

In an AP, with first term ‘a’ and common differenced, the nth term (or the general term) is given by,
an=a+(n-l)d
Note : An AP can be finite or infinite according to as the number of terms are finite or infinite.
If there are m terms in an AP, then am is the last term & is sometimes denoted
by’l’.

4. Sum of the FIRST ‘n’ Terms of an A.P.
(I) The sum of the first n terms of an A.P. is given by:
Sn = n/2-[2a+(n – l)d]
where a is the first term and d is the common difference.
(ii) If l is the last term of the finite A.P. say the nth term, then the sum of all terms of the A.P. is given by,
Sn=-n/2[a+l]
Note : Sum of first n positive integers is given by
Sn = n(n +1)/2

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