CBSE Grade 10 Maths Chapter 5 - Arithmetic Progression.

 Introduction to Arithmetic Progression

  • A sequence is a finite or infinite list of numbers following a certain pattern. 
          For example: 1, 2, 3, 4, 5… ( infinite.sequence of natural numbers)
  • A series is the sum of the elements in the corresponding sequence. 

          For example: 1+2+3+4+5….( series of natural numbers)

  •  Each number in a sequence or a series is called a term.
  • A progression is a sequence in which the general term can be can be expressed using a mathematical formula.

Arithmetic Progression

  • An arithmetic progression (A.P) is a progression in which the difference between two consecutive terms is constant.
  • An Arithmetic Progression is a sequence of numbers in which we get each term by adding a particular number to the previous term, except the first term.
        Example : 

Common Difference

  • The difference between two consecutive terms in an AP, (which is constant) is the “common difference“(d) of an A.P. 

  • In the progression: 2, 5, 8, 11, 14 …the common difference is 3.

  • As it is the difference between any two consecutive terms, for any A.P, if the common difference is:

1)  positive, the AP is increasing.  
     
     Example : for AP = 20,30,40,50 ....., common difference is 10  
 
2)  zero, the AP is constant.
 
    Example : for AP = 9,9,9,9....., common difference is 0  
 
3) negative, the A.P is decreasing.
 
Example : for AP = 20,15,10,5 ....., common difference is -5 

Finite and Infinite AP

  • A finite AP is an A.P in which the number of terms is finite.
          For example: 229, 329, 429, 529, 629
  • An infinite A.P is an A.P in which the number of terms is infinite.

         For example: 2, 4, 6, 8, 10, 12, 14, 16, 18…..…

  • A finite A.P will have the last term, whereas an infinite A.P won’t.

The nth term of an AP

The nth term of an A.P is given by 

where  

an = nth term of AP
a1 = is the first term
d = is a common difference 
n = number of terms.
  
Example

Find the 11th term of the AP: 24, 20, 16,…

Solution

Given a = 24, n = 11, d = 20 – 24 = – 4

an = a + (n - 1)d

a11 = 24 + (11-1) – 4

= 24 + (10) – 4

=24 – 40

= -16

 Arithmetic Series

The arithmetic series is the sum of all the terms of the arithmetic sequence.

The arithmetic series is in the form of

{a + (a + d) + (a + 2d) + (a + 3d) + .........}

 where  

a1 = is the first term
d = is a common difference 

Also, d=0, OR d>0, OR d<0

Example

Given sequence is 2, 5, 8, 11, 14,…

Here, a = 2 and d = 3

d = 5 – 2 = 8 – 5 = 11 – 8 = 3

First term is a = 2

Second term is a + d = 2 + 3 = 5

Third term is a + 2d = 2 + 6 = 8 and so on.

Sum of Terms in an AP

The sum to n terms of an A.P is given by:

 

 where  

Sn = sum of n terms of AP
a1 = is the first term
d = is a common difference 
n = number of terms.

The sum of n terms of an A.P is also given by

 

where,
 
Sn = sum of n terms of AP
a1 = is the first term
l = the last term of the A.P
n = number of terms.

 Also,

last term (l) = a + (n – 1)d

Arithmetic Mean (A.M)

  • The Arithmetic Mean is the simple average of a given set of numbers. 
  • The arithmetic mean of a set of numbers is given by:

Arithmetic Mean (A.M) =Sum of terms/Number of terms

  • The arithmetic mean is defined for any set of numbers. The numbers need not necessarily be in an A.P.

  • Arithmetic mean is the average of the two numbers. If a, b and c are in Arithmetic Progression then the arithmetic mean of a and c will be

    b= (a+c)/2

Basic Adding Patterns in an AP

The sum of two terms that are equidistant from either end of an AP is constant.

For example:  in an A.P: 2,5,8,11,14,17…

T1+T6 = 2+17=19

T2+T5 = 5+14 =19 and so on...

Algebraically, this can be represented as

Tr+T(n-r)+1= constant

Sum of first n natural numbers

The sum of first n natural numbers is given by:

Sn=n(n+1)/2 

This formula is derived by treating the sequence of natural numbers as an A.P where the first term (a) = 1 and the common difference (d) = 1.

 Remark:

  •  The sum of the infinite arithmetic sequence does not exist.

  • The difference between the sum of the first n terms and first (n - 1) terms is also the nth term of the given Arithmetic Progression.

an = Sn – Sn-1