Textbook: Arithmetic Progression | Mathematics Class 10 (Maharashtra SSC Board) PDF Download

 Page 1


55
 · Sequence    · n
th
 term of an A.P. 
 · Arithmetic Progression  · Sum of n terms of an A.P. 
Let’s study.
Let’s learn.
Sequence
 We write numbers 1, 2, 3, 4, . . .  in an order. In this order we can tell the position 
of any number. For example, number 13 is at 13
th
 position. The numbers 1, 4, 9, 16, 
25, 36, 49, . . .  are also written in a particular order. Here 16 = 4
2
 is at 4
th
 position. 
similarly, 25 = 5
2
 is at the 5
th
 position; 49 = 7
2
 is at the 7
th
 position. In this set of 
numbers also, place of each number is detremined. 
 A set of numbers where the numbers are arranged in a definite order, like the 
natural numbers, is called a sequence.  
 In a sequence a particular number is written at a particular position. If the numbers  
are written as a
1
, a
2
, a
3
, a
4
 . . . then a
1
 is first, a
2
 is second, . . .  and so on. It is clear 
that a
n
 is at the n
th
 place. A sequence of the numbers is also represented by alphabets 
f
1
, f
2
, f
3
, . . .  and we find that there is a definite order in which numbers are arranged. 
 When students stand in a row for drill on the playground they form a sequence. 
We have experienced that some sequences have a particular pattern.  
 Complete the given pattern 
Pattern
Number 
of circles 
1 3 5 7
3 Arithmetic Progression
Page 2


55
 · Sequence    · n
th
 term of an A.P. 
 · Arithmetic Progression  · Sum of n terms of an A.P. 
Let’s study.
Let’s learn.
Sequence
 We write numbers 1, 2, 3, 4, . . .  in an order. In this order we can tell the position 
of any number. For example, number 13 is at 13
th
 position. The numbers 1, 4, 9, 16, 
25, 36, 49, . . .  are also written in a particular order. Here 16 = 4
2
 is at 4
th
 position. 
similarly, 25 = 5
2
 is at the 5
th
 position; 49 = 7
2
 is at the 7
th
 position. In this set of 
numbers also, place of each number is detremined. 
 A set of numbers where the numbers are arranged in a definite order, like the 
natural numbers, is called a sequence.  
 In a sequence a particular number is written at a particular position. If the numbers  
are written as a
1
, a
2
, a
3
, a
4
 . . . then a
1
 is first, a
2
 is second, . . .  and so on. It is clear 
that a
n
 is at the n
th
 place. A sequence of the numbers is also represented by alphabets 
f
1
, f
2
, f
3
, . . .  and we find that there is a definite order in which numbers are arranged. 
 When students stand in a row for drill on the playground they form a sequence. 
We have experienced that some sequences have a particular pattern.  
 Complete the given pattern 
Pattern
Number 
of circles 
1 3 5 7
3 Arithmetic Progression
56
Pattern
    
    
 
      
      
      
         
         
         
         
 
Number of 
triangles
5 8 11
 Look at the patterns of the numbers. Try to find a rule to obtain the next number 
from its preceding number. This helps us to write all the next numbers. 
 See the numbers 2, 11, -6, 0, 5, -37, 8, 2, 61 written in this order.
 Here a
1 
= 2, a
2 
= 11, a
3 
= -6, . . .  This list of numbers is also a sequence. But in 
this case we cannot tell why a particular term is at a particular position ; similarly we 
cannot tell a definite relation between the consecutive terms.  
 In general, only those sequences are studied where there is a rule which determines 
the next term. 
 For example  (1) 4, 8, 12, 16 . . .  (2) 2, 4, 8, 16, 32, . . . 
               (3) 
1
5
, 
1
10
, 
1
15
, 
1
20
. . .  
 Terms in a sequence
 In a sequence, ordered terms are represented as t
1
, t
2
, t
3
, . . .  . .t
n
 . . . In general 
sequence is written as {t
n
}. If the sequence is infinite, for every positive integer n, 
there is a term t
n
. 
Activity I : Some sequences are given below. Show the positions of the terms 
by t
1
, t
2
, t
3
, . . . 
 (1) 9, 15, 21, 27, . . .   Here t
1
= 9,  t
2
= 15, t
3
= 21, . . .
 (2) 7, 7, 7, 7, . . .  Here t
1
= 7, t
2
= ,  t
3
= , . . .
 (3) -2, -6, -10, -14, . . .  Here t
1
= -2, t
2
= , t
3
= , . . .
 Activity II : Some sequences are given below. Check whether there is any rule 
among the terms. Find the similarity between two sequences. 
 To check the rule for the terms of the sequence look at the arrangements on the 
next page, and fill the empty boxes suitably.
 (1) 1, 4, 7, 10, 13, . . .      (2) 6, 12, 18, 24, . . . 
 (3) 3, 3, 3, 3, . . .           (4) 4, 16, 64, . . . 
 (5) -1, -1.5, -2, -2.5,  . . .  (6) 1
3
, 2
3
, 3
3
, 4
3
, . . . 
Page 3


55
 · Sequence    · n
th
 term of an A.P. 
 · Arithmetic Progression  · Sum of n terms of an A.P. 
Let’s study.
Let’s learn.
Sequence
 We write numbers 1, 2, 3, 4, . . .  in an order. In this order we can tell the position 
of any number. For example, number 13 is at 13
th
 position. The numbers 1, 4, 9, 16, 
25, 36, 49, . . .  are also written in a particular order. Here 16 = 4
2
 is at 4
th
 position. 
similarly, 25 = 5
2
 is at the 5
th
 position; 49 = 7
2
 is at the 7
th
 position. In this set of 
numbers also, place of each number is detremined. 
 A set of numbers where the numbers are arranged in a definite order, like the 
natural numbers, is called a sequence.  
 In a sequence a particular number is written at a particular position. If the numbers  
are written as a
1
, a
2
, a
3
, a
4
 . . . then a
1
 is first, a
2
 is second, . . .  and so on. It is clear 
that a
n
 is at the n
th
 place. A sequence of the numbers is also represented by alphabets 
f
1
, f
2
, f
3
, . . .  and we find that there is a definite order in which numbers are arranged. 
 When students stand in a row for drill on the playground they form a sequence. 
We have experienced that some sequences have a particular pattern.  
 Complete the given pattern 
Pattern
Number 
of circles 
1 3 5 7
3 Arithmetic Progression
56
Pattern
    
    
 
      
      
      
         
         
         
         
 
Number of 
triangles
5 8 11
 Look at the patterns of the numbers. Try to find a rule to obtain the next number 
from its preceding number. This helps us to write all the next numbers. 
 See the numbers 2, 11, -6, 0, 5, -37, 8, 2, 61 written in this order.
 Here a
1 
= 2, a
2 
= 11, a
3 
= -6, . . .  This list of numbers is also a sequence. But in 
this case we cannot tell why a particular term is at a particular position ; similarly we 
cannot tell a definite relation between the consecutive terms.  
 In general, only those sequences are studied where there is a rule which determines 
the next term. 
 For example  (1) 4, 8, 12, 16 . . .  (2) 2, 4, 8, 16, 32, . . . 
               (3) 
1
5
, 
1
10
, 
1
15
, 
1
20
. . .  
 Terms in a sequence
 In a sequence, ordered terms are represented as t
1
, t
2
, t
3
, . . .  . .t
n
 . . . In general 
sequence is written as {t
n
}. If the sequence is infinite, for every positive integer n, 
there is a term t
n
. 
Activity I : Some sequences are given below. Show the positions of the terms 
by t
1
, t
2
, t
3
, . . . 
 (1) 9, 15, 21, 27, . . .   Here t
1
= 9,  t
2
= 15, t
3
= 21, . . .
 (2) 7, 7, 7, 7, . . .  Here t
1
= 7, t
2
= ,  t
3
= , . . .
 (3) -2, -6, -10, -14, . . .  Here t
1
= -2, t
2
= , t
3
= , . . .
 Activity II : Some sequences are given below. Check whether there is any rule 
among the terms. Find the similarity between two sequences. 
 To check the rule for the terms of the sequence look at the arrangements on the 
next page, and fill the empty boxes suitably.
 (1) 1, 4, 7, 10, 13, . . .      (2) 6, 12, 18, 24, . . . 
 (3) 3, 3, 3, 3, . . .           (4) 4, 16, 64, . . . 
 (5) -1, -1.5, -2, -2.5,  . . .  (6) 1
3
, 2
3
, 3
3
, 4
3
, . . . 
57
 Let’s find the relation in these sequences. Let’s understand the thought behind it.
 (1) 1          4               7            10                         ,  . . . 
        1+3       4+3        7+3          10+3            
 (2)  6         12            18               24                       ,  . . . 
        6 + 6       12 + 6        18 + 6               
 (3) 3         3               3               3                      . . . 
        3 + 0        3 + 0        3 + 0             
 (4) 4         16            64                             , . . . 
         4 ´ 4      16 ´ 4    64 ´ 4      ´ 4           ´ 4
 (5) -1              -1.5                   -2                  -2.5                        . . . 
        (-1)+(-0.5)    -1.5 +(-0.5)    -2 + (-0.5)       -2.5 + (-0.5) 
 (6) 1
3
,    2
3
,     3
3
,    . . .  
 Here in the sequences (1), (2), (3), (5), the similarity is that next term is obtained 
by adding a particular number to the previous number. Each ot these sequences is 
called an Arithmetic Progression. 
 Sequence (4) is not an arithmetic progression. In this sequence the next term 
is obtained by mutliplying the previous term by a particular number. This type of 
sequences is called a Geometric Progression.
 Sequence (6) is neither arithmetic progression nor geometric progression. 
 This year we are going to study arithmetic progression.   
    Arithmetic Progression
 Some sequences are given below. For every sequence write the next three terms. 
    (1) 100, 70, 40, 10, . . . (2) -7, -4, -1, 2, . . .   (3) 4, 4, 4, . . . 
Page 4


55
 · Sequence    · n
th
 term of an A.P. 
 · Arithmetic Progression  · Sum of n terms of an A.P. 
Let’s study.
Let’s learn.
Sequence
 We write numbers 1, 2, 3, 4, . . .  in an order. In this order we can tell the position 
of any number. For example, number 13 is at 13
th
 position. The numbers 1, 4, 9, 16, 
25, 36, 49, . . .  are also written in a particular order. Here 16 = 4
2
 is at 4
th
 position. 
similarly, 25 = 5
2
 is at the 5
th
 position; 49 = 7
2
 is at the 7
th
 position. In this set of 
numbers also, place of each number is detremined. 
 A set of numbers where the numbers are arranged in a definite order, like the 
natural numbers, is called a sequence.  
 In a sequence a particular number is written at a particular position. If the numbers  
are written as a
1
, a
2
, a
3
, a
4
 . . . then a
1
 is first, a
2
 is second, . . .  and so on. It is clear 
that a
n
 is at the n
th
 place. A sequence of the numbers is also represented by alphabets 
f
1
, f
2
, f
3
, . . .  and we find that there is a definite order in which numbers are arranged. 
 When students stand in a row for drill on the playground they form a sequence. 
We have experienced that some sequences have a particular pattern.  
 Complete the given pattern 
Pattern
Number 
of circles 
1 3 5 7
3 Arithmetic Progression
56
Pattern
    
    
 
      
      
      
         
         
         
         
 
Number of 
triangles
5 8 11
 Look at the patterns of the numbers. Try to find a rule to obtain the next number 
from its preceding number. This helps us to write all the next numbers. 
 See the numbers 2, 11, -6, 0, 5, -37, 8, 2, 61 written in this order.
 Here a
1 
= 2, a
2 
= 11, a
3 
= -6, . . .  This list of numbers is also a sequence. But in 
this case we cannot tell why a particular term is at a particular position ; similarly we 
cannot tell a definite relation between the consecutive terms.  
 In general, only those sequences are studied where there is a rule which determines 
the next term. 
 For example  (1) 4, 8, 12, 16 . . .  (2) 2, 4, 8, 16, 32, . . . 
               (3) 
1
5
, 
1
10
, 
1
15
, 
1
20
. . .  
 Terms in a sequence
 In a sequence, ordered terms are represented as t
1
, t
2
, t
3
, . . .  . .t
n
 . . . In general 
sequence is written as {t
n
}. If the sequence is infinite, for every positive integer n, 
there is a term t
n
. 
Activity I : Some sequences are given below. Show the positions of the terms 
by t
1
, t
2
, t
3
, . . . 
 (1) 9, 15, 21, 27, . . .   Here t
1
= 9,  t
2
= 15, t
3
= 21, . . .
 (2) 7, 7, 7, 7, . . .  Here t
1
= 7, t
2
= ,  t
3
= , . . .
 (3) -2, -6, -10, -14, . . .  Here t
1
= -2, t
2
= , t
3
= , . . .
 Activity II : Some sequences are given below. Check whether there is any rule 
among the terms. Find the similarity between two sequences. 
 To check the rule for the terms of the sequence look at the arrangements on the 
next page, and fill the empty boxes suitably.
 (1) 1, 4, 7, 10, 13, . . .      (2) 6, 12, 18, 24, . . . 
 (3) 3, 3, 3, 3, . . .           (4) 4, 16, 64, . . . 
 (5) -1, -1.5, -2, -2.5,  . . .  (6) 1
3
, 2
3
, 3
3
, 4
3
, . . . 
57
 Let’s find the relation in these sequences. Let’s understand the thought behind it.
 (1) 1          4               7            10                         ,  . . . 
        1+3       4+3        7+3          10+3            
 (2)  6         12            18               24                       ,  . . . 
        6 + 6       12 + 6        18 + 6               
 (3) 3         3               3               3                      . . . 
        3 + 0        3 + 0        3 + 0             
 (4) 4         16            64                             , . . . 
         4 ´ 4      16 ´ 4    64 ´ 4      ´ 4           ´ 4
 (5) -1              -1.5                   -2                  -2.5                        . . . 
        (-1)+(-0.5)    -1.5 +(-0.5)    -2 + (-0.5)       -2.5 + (-0.5) 
 (6) 1
3
,    2
3
,     3
3
,    . . .  
 Here in the sequences (1), (2), (3), (5), the similarity is that next term is obtained 
by adding a particular number to the previous number. Each ot these sequences is 
called an Arithmetic Progression. 
 Sequence (4) is not an arithmetic progression. In this sequence the next term 
is obtained by mutliplying the previous term by a particular number. This type of 
sequences is called a Geometric Progression.
 Sequence (6) is neither arithmetic progression nor geometric progression. 
 This year we are going to study arithmetic progression.   
    Arithmetic Progression
 Some sequences are given below. For every sequence write the next three terms. 
    (1) 100, 70, 40, 10, . . . (2) -7, -4, -1, 2, . . .   (3) 4, 4, 4, . . . 
58
 In the given sequences, observe how the next term is obtained. 
 (1) 100           70                40        10                 
                100+(-30)        70+(-30)        40+(-30)     10+(-30)   (-20)+(-30)     
 (2) -7         -4              -1         2                        
       
           -7+3              -4+3            -1+3           2+3                5+3 
 (3) 4           4              4              4       4 . . .  .
        
        4 + 0            4 + 0               4 + 0           4 + 0
 In each sequence above, every term is obtained by adding a particular number in  
 the previous term. The difference between two consecutive terms is constant. 
 The diference in ex. (i) is negative, in ex. (ii) it is positive and in ex. (iii) it is zero. 
 If the difference between two consecutive terms is constant then it is called the  
 common difference and is generally denoted by letter d.
 In the given sequence if the difference between two consecutive terms (t
n +1
- t
n
)
 
is  
 constant then the sequence is called Arithmetic Progression (A.P.). In this sequence 
  t
n +1
- t
n 
= d is the common difference.  
 In an A.P. if first term is denoted by a and common difference is d then, 
 t
1
 = a ,   t
2
= a + d
 t
3
= (a + d) + d = a + 2d
 A.P. having first term as a and common difference d  is  
 a, (a + d), (a + 2d), (a + 3d), . . . . . . 
Let’s see some examples of A.P. 
Ex.(1) Arifa saved ` 100 every month. In one year the total amount saved after every 
month is as given below.
Month
I II III IV V VI VII VIII IX X XI XII
Saving (’) 100 200 300 400 500 600 700 800 900 1000 1100 1200
 The numbers showing the total saving after every month are in A.P. 
3
rd
 term 2
nd
 term 1
st
 term
-50
  5
-20
  8
Page 5


55
 · Sequence    · n
th
 term of an A.P. 
 · Arithmetic Progression  · Sum of n terms of an A.P. 
Let’s study.
Let’s learn.
Sequence
 We write numbers 1, 2, 3, 4, . . .  in an order. In this order we can tell the position 
of any number. For example, number 13 is at 13
th
 position. The numbers 1, 4, 9, 16, 
25, 36, 49, . . .  are also written in a particular order. Here 16 = 4
2
 is at 4
th
 position. 
similarly, 25 = 5
2
 is at the 5
th
 position; 49 = 7
2
 is at the 7
th
 position. In this set of 
numbers also, place of each number is detremined. 
 A set of numbers where the numbers are arranged in a definite order, like the 
natural numbers, is called a sequence.  
 In a sequence a particular number is written at a particular position. If the numbers  
are written as a
1
, a
2
, a
3
, a
4
 . . . then a
1
 is first, a
2
 is second, . . .  and so on. It is clear 
that a
n
 is at the n
th
 place. A sequence of the numbers is also represented by alphabets 
f
1
, f
2
, f
3
, . . .  and we find that there is a definite order in which numbers are arranged. 
 When students stand in a row for drill on the playground they form a sequence. 
We have experienced that some sequences have a particular pattern.  
 Complete the given pattern 
Pattern
Number 
of circles 
1 3 5 7
3 Arithmetic Progression
56
Pattern
    
    
 
      
      
      
         
         
         
         
 
Number of 
triangles
5 8 11
 Look at the patterns of the numbers. Try to find a rule to obtain the next number 
from its preceding number. This helps us to write all the next numbers. 
 See the numbers 2, 11, -6, 0, 5, -37, 8, 2, 61 written in this order.
 Here a
1 
= 2, a
2 
= 11, a
3 
= -6, . . .  This list of numbers is also a sequence. But in 
this case we cannot tell why a particular term is at a particular position ; similarly we 
cannot tell a definite relation between the consecutive terms.  
 In general, only those sequences are studied where there is a rule which determines 
the next term. 
 For example  (1) 4, 8, 12, 16 . . .  (2) 2, 4, 8, 16, 32, . . . 
               (3) 
1
5
, 
1
10
, 
1
15
, 
1
20
. . .  
 Terms in a sequence
 In a sequence, ordered terms are represented as t
1
, t
2
, t
3
, . . .  . .t
n
 . . . In general 
sequence is written as {t
n
}. If the sequence is infinite, for every positive integer n, 
there is a term t
n
. 
Activity I : Some sequences are given below. Show the positions of the terms 
by t
1
, t
2
, t
3
, . . . 
 (1) 9, 15, 21, 27, . . .   Here t
1
= 9,  t
2
= 15, t
3
= 21, . . .
 (2) 7, 7, 7, 7, . . .  Here t
1
= 7, t
2
= ,  t
3
= , . . .
 (3) -2, -6, -10, -14, . . .  Here t
1
= -2, t
2
= , t
3
= , . . .
 Activity II : Some sequences are given below. Check whether there is any rule 
among the terms. Find the similarity between two sequences. 
 To check the rule for the terms of the sequence look at the arrangements on the 
next page, and fill the empty boxes suitably.
 (1) 1, 4, 7, 10, 13, . . .      (2) 6, 12, 18, 24, . . . 
 (3) 3, 3, 3, 3, . . .           (4) 4, 16, 64, . . . 
 (5) -1, -1.5, -2, -2.5,  . . .  (6) 1
3
, 2
3
, 3
3
, 4
3
, . . . 
57
 Let’s find the relation in these sequences. Let’s understand the thought behind it.
 (1) 1          4               7            10                         ,  . . . 
        1+3       4+3        7+3          10+3            
 (2)  6         12            18               24                       ,  . . . 
        6 + 6       12 + 6        18 + 6               
 (3) 3         3               3               3                      . . . 
        3 + 0        3 + 0        3 + 0             
 (4) 4         16            64                             , . . . 
         4 ´ 4      16 ´ 4    64 ´ 4      ´ 4           ´ 4
 (5) -1              -1.5                   -2                  -2.5                        . . . 
        (-1)+(-0.5)    -1.5 +(-0.5)    -2 + (-0.5)       -2.5 + (-0.5) 
 (6) 1
3
,    2
3
,     3
3
,    . . .  
 Here in the sequences (1), (2), (3), (5), the similarity is that next term is obtained 
by adding a particular number to the previous number. Each ot these sequences is 
called an Arithmetic Progression. 
 Sequence (4) is not an arithmetic progression. In this sequence the next term 
is obtained by mutliplying the previous term by a particular number. This type of 
sequences is called a Geometric Progression.
 Sequence (6) is neither arithmetic progression nor geometric progression. 
 This year we are going to study arithmetic progression.   
    Arithmetic Progression
 Some sequences are given below. For every sequence write the next three terms. 
    (1) 100, 70, 40, 10, . . . (2) -7, -4, -1, 2, . . .   (3) 4, 4, 4, . . . 
58
 In the given sequences, observe how the next term is obtained. 
 (1) 100           70                40        10                 
                100+(-30)        70+(-30)        40+(-30)     10+(-30)   (-20)+(-30)     
 (2) -7         -4              -1         2                        
       
           -7+3              -4+3            -1+3           2+3                5+3 
 (3) 4           4              4              4       4 . . .  .
        
        4 + 0            4 + 0               4 + 0           4 + 0
 In each sequence above, every term is obtained by adding a particular number in  
 the previous term. The difference between two consecutive terms is constant. 
 The diference in ex. (i) is negative, in ex. (ii) it is positive and in ex. (iii) it is zero. 
 If the difference between two consecutive terms is constant then it is called the  
 common difference and is generally denoted by letter d.
 In the given sequence if the difference between two consecutive terms (t
n +1
- t
n
)
 
is  
 constant then the sequence is called Arithmetic Progression (A.P.). In this sequence 
  t
n +1
- t
n 
= d is the common difference.  
 In an A.P. if first term is denoted by a and common difference is d then, 
 t
1
 = a ,   t
2
= a + d
 t
3
= (a + d) + d = a + 2d
 A.P. having first term as a and common difference d  is  
 a, (a + d), (a + 2d), (a + 3d), . . . . . . 
Let’s see some examples of A.P. 
Ex.(1) Arifa saved ` 100 every month. In one year the total amount saved after every 
month is as given below.
Month
I II III IV V VI VII VIII IX X XI XII
Saving (’) 100 200 300 400 500 600 700 800 900 1000 1100 1200
 The numbers showing the total saving after every month are in A.P. 
3
rd
 term 2
nd
 term 1
st
 term
-50
  5
-20
  8
59
(1) In a sequence if difference (t
n +1
- t
n
)
 
is constant then the sequence 
 is called an arithmetic progression. 
(2) In an A.P. the difference between two consecutive terms is constant 
 and is denoted by d. 
(3) Difference d can be positive, negative or zero. 
(4) In an A.P. if the first term is a, and common difference is d then the 
 terms in the sequence are a, (a + d), (a + 2d),   . . . 
Ex. (2) Pranav borrowed ` 10000 from his friend and agreed to repay ` 1000 per month. So 
the remaining amount to be paid in every month will be as follows. 
No. of month
1 2 3 4 5 ... ... ...
Amount to be 
paid (`)
10,000 9,000 8,000 7,000 ... 2,000 1,000 0
Ex. (3) Consider the table of 5, that is numbers divisible by 5. 
 5, 10, 15, 20, . . .  50, 55, 60, . . .  . . is an inifinite A.P. 
 Ex (1) and (2) are finite A.P. while (3) is an infinite A.P. 
Activity : Write one example of finite and infinite A.P. each. 
ÒÒÒ? Solved examples ÒÒÒ
Ex. (1) Which of the following sequences are A.P ? If it is an A.P, find next two terms. 
 (i) 5, 12, 19, 26, . . .     (ii) 2, -2, -6, -10, . . . 
 (iii) 1, 1, 2, 2, 3, 3, . . .  (iv) 
3
2
, 
1
2
, -
1
2
, . . . 
Solution :  (i) In this sequence 5, 12, 19, 26, . . . , 
   First term = t
1
= 5,    t
2
= 12,   t
3
= 19, . . . 
    t
2
- t
1
= 12 - 5 = 7
    t
3
- t
2
= 19 - 12 = 7
  Here first term is 5 and common difference which is constant is d = 7 
  \ This sequence is an A.P. 
   Next two terms in this A.P. are 26 + 7 = 33  and  33 + 7 = 40.
   Next two terms in given A.P. are 33 and 40
Let’s remember!