Detailed Notes: Sequences and Series | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Sequence & Series 
Sequence: 
A sequence is a function whose domain is the set ?? of natural numbers. Since the domain 
for every sequence is the set N of natural numbers, therefore a sequence is represented by 
its range. If ?? : ?? ? ?? , then ?? ( ?? ) = ?? ?? , ?? ? ?? is called a sequence and is denoted by 
{?? ( 1) , ?? ( 2) , ?? ( 3) , … … … … … . } =. {?? 1
, ?? 2
, t
3
, … … … … … . . } = {?? ?? } 
Real Sequence: 
A sequence whose range is a subset of ?? is called a real sequence. 
 
Types of sequence: 
On the basis of the number of terms there are two types of sequence. 
(i) Finite sequences: A sequence is said to be finite if it has finite number of terms. 
(ii) Infinite sequences: A sequence is said to be infinite if it has infinitely many terms. 
Problem 1: Write down the sequence whose n
th 
 term is 
( -2)
n
( -1)
n
+2
 
Solution:   Let ?? ?? =
( -2)
?? ( -1)
?? +2
 
put ?? = 1,2,3,4, … … … … …. we get 
t
1
= -2, t
2
=
4
3
, t
3
= -8, t
4
=
16
3
 
so the sequence is -2, , -8,
16
3
 
Series: 
By adding or subtracting the terms of a sequence, we get an expression which is called a 
series. If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? is a sequence, then the expression ?? 1
+ ?? 2
+ ?? 3
+ ? . . +?? ?? is a 
series. 
e.g. 
Page 2


Sequence & Series 
Sequence: 
A sequence is a function whose domain is the set ?? of natural numbers. Since the domain 
for every sequence is the set N of natural numbers, therefore a sequence is represented by 
its range. If ?? : ?? ? ?? , then ?? ( ?? ) = ?? ?? , ?? ? ?? is called a sequence and is denoted by 
{?? ( 1) , ?? ( 2) , ?? ( 3) , … … … … … . } =. {?? 1
, ?? 2
, t
3
, … … … … … . . } = {?? ?? } 
Real Sequence: 
A sequence whose range is a subset of ?? is called a real sequence. 
 
Types of sequence: 
On the basis of the number of terms there are two types of sequence. 
(i) Finite sequences: A sequence is said to be finite if it has finite number of terms. 
(ii) Infinite sequences: A sequence is said to be infinite if it has infinitely many terms. 
Problem 1: Write down the sequence whose n
th 
 term is 
( -2)
n
( -1)
n
+2
 
Solution:   Let ?? ?? =
( -2)
?? ( -1)
?? +2
 
put ?? = 1,2,3,4, … … … … …. we get 
t
1
= -2, t
2
=
4
3
, t
3
= -8, t
4
=
16
3
 
so the sequence is -2, , -8,
16
3
 
Series: 
By adding or subtracting the terms of a sequence, we get an expression which is called a 
series. If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? is a sequence, then the expression ?? 1
+ ?? 2
+ ?? 3
+ ? . . +?? ?? is a 
series. 
e.g. 
(i) 1 + 2 + 3 + 4 + ? … … … … . . ?? 
(iii) -1 + 3 - 9 + 27 - 
Progression: 
The word progression refers to sequence or series - finite or infinite 
Arithmetic progression (A.P.): 
A.P. is a sequence whose successive terms are obtained by adding a fixed number 'd' to the 
preceding terms. This fixed number 'd' is called the common difference. If ?? is the first 
term & ?? the common difference, then A.P. can be written as , ?? + ?? , ?? + 2?? , ?? + ( ?? -
1) ?? , … …. 
e.g. -4, -1,2,5   
?? th 
 term of an A.P.: 
Let 'a' be the first term and 'd' be the common difference of an A.P., then ?? ?? = ?? + ( ?? -
1) ?? , where ?? = ?? ?? - ?? ?? -1
 
Problem 2: Find the number of terms in the sequence 4, 7, 10, 13, .82. 
Solution: Let a be the first term and ?? be the common difference 
?? = 4, ?? = 3  so  82 = 4 + ( ?? - 1) 3
 ? ?? = 27
 
The sum of first ?? terms of an A.P.: 
If ?? is first term and ?? is common difference, then sum of the first ?? terms of ???? is 
?? ?? =
?? 2
[2?? + ( ?? - 1) ?? ] 
=
n
2
[a + l] = nt(
n+1
2
), for n is odd. (Where l is the last term and t
(
n+1
2
)
 is the middle term.) 
Note: For any sequence {?? ?? }, whose sum of first ?? terms is ?? ?? , ?? th 
 term, ?? ?? = ?? ?? - ?? ?? -1
. 
Problem 3: If in an A.P., 3rd term is 18 and 7 term is 30 , then find sum of its first 17 terms 
Solution: Let a be the first term and d be the common difference 
?? + 2?? = 18 
?? + 6?? = 30 
Page 3


Sequence & Series 
Sequence: 
A sequence is a function whose domain is the set ?? of natural numbers. Since the domain 
for every sequence is the set N of natural numbers, therefore a sequence is represented by 
its range. If ?? : ?? ? ?? , then ?? ( ?? ) = ?? ?? , ?? ? ?? is called a sequence and is denoted by 
{?? ( 1) , ?? ( 2) , ?? ( 3) , … … … … … . } =. {?? 1
, ?? 2
, t
3
, … … … … … . . } = {?? ?? } 
Real Sequence: 
A sequence whose range is a subset of ?? is called a real sequence. 
 
Types of sequence: 
On the basis of the number of terms there are two types of sequence. 
(i) Finite sequences: A sequence is said to be finite if it has finite number of terms. 
(ii) Infinite sequences: A sequence is said to be infinite if it has infinitely many terms. 
Problem 1: Write down the sequence whose n
th 
 term is 
( -2)
n
( -1)
n
+2
 
Solution:   Let ?? ?? =
( -2)
?? ( -1)
?? +2
 
put ?? = 1,2,3,4, … … … … …. we get 
t
1
= -2, t
2
=
4
3
, t
3
= -8, t
4
=
16
3
 
so the sequence is -2, , -8,
16
3
 
Series: 
By adding or subtracting the terms of a sequence, we get an expression which is called a 
series. If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? is a sequence, then the expression ?? 1
+ ?? 2
+ ?? 3
+ ? . . +?? ?? is a 
series. 
e.g. 
(i) 1 + 2 + 3 + 4 + ? … … … … . . ?? 
(iii) -1 + 3 - 9 + 27 - 
Progression: 
The word progression refers to sequence or series - finite or infinite 
Arithmetic progression (A.P.): 
A.P. is a sequence whose successive terms are obtained by adding a fixed number 'd' to the 
preceding terms. This fixed number 'd' is called the common difference. If ?? is the first 
term & ?? the common difference, then A.P. can be written as , ?? + ?? , ?? + 2?? , ?? + ( ?? -
1) ?? , … …. 
e.g. -4, -1,2,5   
?? th 
 term of an A.P.: 
Let 'a' be the first term and 'd' be the common difference of an A.P., then ?? ?? = ?? + ( ?? -
1) ?? , where ?? = ?? ?? - ?? ?? -1
 
Problem 2: Find the number of terms in the sequence 4, 7, 10, 13, .82. 
Solution: Let a be the first term and ?? be the common difference 
?? = 4, ?? = 3  so  82 = 4 + ( ?? - 1) 3
 ? ?? = 27
 
The sum of first ?? terms of an A.P.: 
If ?? is first term and ?? is common difference, then sum of the first ?? terms of ???? is 
?? ?? =
?? 2
[2?? + ( ?? - 1) ?? ] 
=
n
2
[a + l] = nt(
n+1
2
), for n is odd. (Where l is the last term and t
(
n+1
2
)
 is the middle term.) 
Note: For any sequence {?? ?? }, whose sum of first ?? terms is ?? ?? , ?? th 
 term, ?? ?? = ?? ?? - ?? ?? -1
. 
Problem 3: If in an A.P., 3rd term is 18 and 7 term is 30 , then find sum of its first 17 terms 
Solution: Let a be the first term and d be the common difference 
?? + 2?? = 18 
?? + 6?? = 30 
?? = 3, ?? = 12 
?? 17
=
17
2
[2 × 12 + 16 × 3] = 612 
Problem 4: Find the sum of all odd numbers between 1 and 1000 which are divisible by 3 
Solution:   Odd numbers between 1 and 1000 are 
3,5,7,9,11,13, - - - - -993,995,997,999. 
Those numbers which are divisible by 3 are 
3,9,15,21 993, 999 
They form an A.P. of which ?? = 3, ?? = 6, l = 999 ? ?? = 167 
?? =
n
2
[a + l] = 83667 
Problem 5: The ratio between the sum of ?? term of two A.P.'s is 3?? + 8: 7?? + 15. Then find 
the ratio between their 12 th term 
Solution: 
?? ?? ?? ?? 
'
=
( ?? /2) [2?? + ( ?? - 1) ?? ]
( ?? /2) [2?? '
+ ( ?? - 1) ?? '
]
=
3?? + 8
7?? + 15
 or 
?? + {( ?? - 1) /2}?? ?? '
+ ( ?? - 1) /2?? '
=
3?? + 8
7?? + 15
( ?? ) 
we have to find 
?? 12
?? 12
=
?? +11?? ?? '
+11?? '
 
choosing ( ?? - 1) /2 = 11 or ?? = 23 in (1), 
we get 
?? 12
 T
12
 
'
=
?? +11?? ?? '
+11?? '
=
3( 23) +8
( 23) ×7+15
=
77
176
=
7
16
 
Problem 6: If sum of ?? terms of a sequence is given by ?? ?? = 3?? 2
- 4?? , find its 50
th 
 term. 
Solution: Let ?? ?? is ?? th 
 term of the sequence so ?? ?? = ?? ?? - ?? ?? -1
. 
 = 3?? 2
- 4?? - 3( ?? - 1)
2
+ 4( ?? - 1) = 6?? - 7
 so  ?? 50
= 293
 
Remarks: 
(i) The first term and common difference can be zero, positive or negative (or any complex 
number.) 
(ii) If ?? , ?? , ?? are in A.P. ? 2?? = ?? + ?? & if ?? , ?? , ?? , ?? are in A.P. ? ?? + ?? = ?? + ?? . 
Page 4


Sequence & Series 
Sequence: 
A sequence is a function whose domain is the set ?? of natural numbers. Since the domain 
for every sequence is the set N of natural numbers, therefore a sequence is represented by 
its range. If ?? : ?? ? ?? , then ?? ( ?? ) = ?? ?? , ?? ? ?? is called a sequence and is denoted by 
{?? ( 1) , ?? ( 2) , ?? ( 3) , … … … … … . } =. {?? 1
, ?? 2
, t
3
, … … … … … . . } = {?? ?? } 
Real Sequence: 
A sequence whose range is a subset of ?? is called a real sequence. 
 
Types of sequence: 
On the basis of the number of terms there are two types of sequence. 
(i) Finite sequences: A sequence is said to be finite if it has finite number of terms. 
(ii) Infinite sequences: A sequence is said to be infinite if it has infinitely many terms. 
Problem 1: Write down the sequence whose n
th 
 term is 
( -2)
n
( -1)
n
+2
 
Solution:   Let ?? ?? =
( -2)
?? ( -1)
?? +2
 
put ?? = 1,2,3,4, … … … … …. we get 
t
1
= -2, t
2
=
4
3
, t
3
= -8, t
4
=
16
3
 
so the sequence is -2, , -8,
16
3
 
Series: 
By adding or subtracting the terms of a sequence, we get an expression which is called a 
series. If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? is a sequence, then the expression ?? 1
+ ?? 2
+ ?? 3
+ ? . . +?? ?? is a 
series. 
e.g. 
(i) 1 + 2 + 3 + 4 + ? … … … … . . ?? 
(iii) -1 + 3 - 9 + 27 - 
Progression: 
The word progression refers to sequence or series - finite or infinite 
Arithmetic progression (A.P.): 
A.P. is a sequence whose successive terms are obtained by adding a fixed number 'd' to the 
preceding terms. This fixed number 'd' is called the common difference. If ?? is the first 
term & ?? the common difference, then A.P. can be written as , ?? + ?? , ?? + 2?? , ?? + ( ?? -
1) ?? , … …. 
e.g. -4, -1,2,5   
?? th 
 term of an A.P.: 
Let 'a' be the first term and 'd' be the common difference of an A.P., then ?? ?? = ?? + ( ?? -
1) ?? , where ?? = ?? ?? - ?? ?? -1
 
Problem 2: Find the number of terms in the sequence 4, 7, 10, 13, .82. 
Solution: Let a be the first term and ?? be the common difference 
?? = 4, ?? = 3  so  82 = 4 + ( ?? - 1) 3
 ? ?? = 27
 
The sum of first ?? terms of an A.P.: 
If ?? is first term and ?? is common difference, then sum of the first ?? terms of ???? is 
?? ?? =
?? 2
[2?? + ( ?? - 1) ?? ] 
=
n
2
[a + l] = nt(
n+1
2
), for n is odd. (Where l is the last term and t
(
n+1
2
)
 is the middle term.) 
Note: For any sequence {?? ?? }, whose sum of first ?? terms is ?? ?? , ?? th 
 term, ?? ?? = ?? ?? - ?? ?? -1
. 
Problem 3: If in an A.P., 3rd term is 18 and 7 term is 30 , then find sum of its first 17 terms 
Solution: Let a be the first term and d be the common difference 
?? + 2?? = 18 
?? + 6?? = 30 
?? = 3, ?? = 12 
?? 17
=
17
2
[2 × 12 + 16 × 3] = 612 
Problem 4: Find the sum of all odd numbers between 1 and 1000 which are divisible by 3 
Solution:   Odd numbers between 1 and 1000 are 
3,5,7,9,11,13, - - - - -993,995,997,999. 
Those numbers which are divisible by 3 are 
3,9,15,21 993, 999 
They form an A.P. of which ?? = 3, ?? = 6, l = 999 ? ?? = 167 
?? =
n
2
[a + l] = 83667 
Problem 5: The ratio between the sum of ?? term of two A.P.'s is 3?? + 8: 7?? + 15. Then find 
the ratio between their 12 th term 
Solution: 
?? ?? ?? ?? 
'
=
( ?? /2) [2?? + ( ?? - 1) ?? ]
( ?? /2) [2?? '
+ ( ?? - 1) ?? '
]
=
3?? + 8
7?? + 15
 or 
?? + {( ?? - 1) /2}?? ?? '
+ ( ?? - 1) /2?? '
=
3?? + 8
7?? + 15
( ?? ) 
we have to find 
?? 12
?? 12
=
?? +11?? ?? '
+11?? '
 
choosing ( ?? - 1) /2 = 11 or ?? = 23 in (1), 
we get 
?? 12
 T
12
 
'
=
?? +11?? ?? '
+11?? '
=
3( 23) +8
( 23) ×7+15
=
77
176
=
7
16
 
Problem 6: If sum of ?? terms of a sequence is given by ?? ?? = 3?? 2
- 4?? , find its 50
th 
 term. 
Solution: Let ?? ?? is ?? th 
 term of the sequence so ?? ?? = ?? ?? - ?? ?? -1
. 
 = 3?? 2
- 4?? - 3( ?? - 1)
2
+ 4( ?? - 1) = 6?? - 7
 so  ?? 50
= 293
 
Remarks: 
(i) The first term and common difference can be zero, positive or negative (or any complex 
number.) 
(ii) If ?? , ?? , ?? are in A.P. ? 2?? = ?? + ?? & if ?? , ?? , ?? , ?? are in A.P. ? ?? + ?? = ?? + ?? . 
(iii) Three numbers in A.P. can be taken as a - d, a, a + d; four numbers in A.P. can be 
taken as a -3?? , ?? - ?? , ?? + ?? , ?? + 3?? ; five numbers in A.P. are ?? - 2?? , ?? - ?? , ?? , ?? + ?? , ?? + 2?? ; 
six terms in A.P. are ?? - 5?? , ?? - 3?? , ?? - ?? , ?? + ?? , ?? + 3?? , ?? + 5?? etc. 
(iv) The sum of the terms of an A.P. equidistant from the beginning & end is constant and 
equal to the sum of first & last terms. 
(v) Any term of an A.P. (except the first) is equal to half the sum of terms which are 
equidistant from it. ?? ?? = 1/2( ?? ?? -?? + ?? ?? +?? ) , ?? < ?? . For ?? = 1, ?? ?? = ( 1/2) ( ?? ?? -1
+ ?? ?? +1
) ; 
For k = 2, ?? ?? = ( 1/2) ( ?? ?? -2
+ ?? ?? +2
) and so on. 
(vi) If each term of an A.P. is increased, decreased, multiplied or divided by the same non-
zero number, then the resulting sequence is also an AP. 
(vii) The sum and difference of two AP's is an AP. 
Problem 7: The numbers t( t
2
+ 1) , -
t
2
2
 and 6 are three consecutive terms of an A.P. If t be 
real, then find the the next two term of A.P. 
Solution:  2?? = ?? + ?? ? - ?? 2
= ?? 3
+ ?? + 6 
or ?? 3
+ ?? 2
+ ?? + 6 = 0 
or  ( ?? + 2) ( ?? 2
- ?? + 3) = 0 ? ?? 2
- ?? + 3 ? 0 ? ?? = -2 
the given numbers are -10, -2,6 
which are in an A.P. with ?? = 8. The next two numbers are 14,22 
Problem 8: If ?? 1
, ?? 2
, ?? 3
, ?? 4
, ?? 5
 are in A.P. with common difference ? 0, then find the value 
of ?
?? =1
5
??? ?? , when ?? 3
= 2. 
Solution:   As ?? 1
, ?? 2
, ?? 3
, ?? 4
, ?? 5
 are in A.P., we have ?? 1
+ ?? 5
= ?? 2
+ ?? 4
= 2?? 3
. Hence ?
?? =1
5
??? ?? =
10. 
Problem 9: If ?? ( ?? + ?? ) , ?? ( ?? + ?? ) , ?? ( ?? + ?? ) are in A.P., prove that 
1
?? ,
1
?? ,
1
?? are also in A.P. 
Solution:  ? ?? ( ?? + ?? ) , ?? ( ?? + ?? ) , ?? ( ?? + ?? ) are in A.P. ?  subtract ???? + ???? + ???? from each 
-???? , -???? , -???? are in A.P. divide by -?????? 
1
a
,
1
 b
,
1
c
 are in A.P.  
Problem 10: If 
?? +?? 1-????
, ?? ,
?? +?? 1-????
 are in A.P. then prove that 
1
?? , ?? 1
?? are in A.P. 
Solution:  ? 
?? +?? 1-????
, ?? ,
?? +?? 1-????
 are in A.P. 
Page 5


Sequence & Series 
Sequence: 
A sequence is a function whose domain is the set ?? of natural numbers. Since the domain 
for every sequence is the set N of natural numbers, therefore a sequence is represented by 
its range. If ?? : ?? ? ?? , then ?? ( ?? ) = ?? ?? , ?? ? ?? is called a sequence and is denoted by 
{?? ( 1) , ?? ( 2) , ?? ( 3) , … … … … … . } =. {?? 1
, ?? 2
, t
3
, … … … … … . . } = {?? ?? } 
Real Sequence: 
A sequence whose range is a subset of ?? is called a real sequence. 
 
Types of sequence: 
On the basis of the number of terms there are two types of sequence. 
(i) Finite sequences: A sequence is said to be finite if it has finite number of terms. 
(ii) Infinite sequences: A sequence is said to be infinite if it has infinitely many terms. 
Problem 1: Write down the sequence whose n
th 
 term is 
( -2)
n
( -1)
n
+2
 
Solution:   Let ?? ?? =
( -2)
?? ( -1)
?? +2
 
put ?? = 1,2,3,4, … … … … …. we get 
t
1
= -2, t
2
=
4
3
, t
3
= -8, t
4
=
16
3
 
so the sequence is -2, , -8,
16
3
 
Series: 
By adding or subtracting the terms of a sequence, we get an expression which is called a 
series. If ?? 1
, ?? 2
, ?? 3
, … . . . ?? ?? is a sequence, then the expression ?? 1
+ ?? 2
+ ?? 3
+ ? . . +?? ?? is a 
series. 
e.g. 
(i) 1 + 2 + 3 + 4 + ? … … … … . . ?? 
(iii) -1 + 3 - 9 + 27 - 
Progression: 
The word progression refers to sequence or series - finite or infinite 
Arithmetic progression (A.P.): 
A.P. is a sequence whose successive terms are obtained by adding a fixed number 'd' to the 
preceding terms. This fixed number 'd' is called the common difference. If ?? is the first 
term & ?? the common difference, then A.P. can be written as , ?? + ?? , ?? + 2?? , ?? + ( ?? -
1) ?? , … …. 
e.g. -4, -1,2,5   
?? th 
 term of an A.P.: 
Let 'a' be the first term and 'd' be the common difference of an A.P., then ?? ?? = ?? + ( ?? -
1) ?? , where ?? = ?? ?? - ?? ?? -1
 
Problem 2: Find the number of terms in the sequence 4, 7, 10, 13, .82. 
Solution: Let a be the first term and ?? be the common difference 
?? = 4, ?? = 3  so  82 = 4 + ( ?? - 1) 3
 ? ?? = 27
 
The sum of first ?? terms of an A.P.: 
If ?? is first term and ?? is common difference, then sum of the first ?? terms of ???? is 
?? ?? =
?? 2
[2?? + ( ?? - 1) ?? ] 
=
n
2
[a + l] = nt(
n+1
2
), for n is odd. (Where l is the last term and t
(
n+1
2
)
 is the middle term.) 
Note: For any sequence {?? ?? }, whose sum of first ?? terms is ?? ?? , ?? th 
 term, ?? ?? = ?? ?? - ?? ?? -1
. 
Problem 3: If in an A.P., 3rd term is 18 and 7 term is 30 , then find sum of its first 17 terms 
Solution: Let a be the first term and d be the common difference 
?? + 2?? = 18 
?? + 6?? = 30 
?? = 3, ?? = 12 
?? 17
=
17
2
[2 × 12 + 16 × 3] = 612 
Problem 4: Find the sum of all odd numbers between 1 and 1000 which are divisible by 3 
Solution:   Odd numbers between 1 and 1000 are 
3,5,7,9,11,13, - - - - -993,995,997,999. 
Those numbers which are divisible by 3 are 
3,9,15,21 993, 999 
They form an A.P. of which ?? = 3, ?? = 6, l = 999 ? ?? = 167 
?? =
n
2
[a + l] = 83667 
Problem 5: The ratio between the sum of ?? term of two A.P.'s is 3?? + 8: 7?? + 15. Then find 
the ratio between their 12 th term 
Solution: 
?? ?? ?? ?? 
'
=
( ?? /2) [2?? + ( ?? - 1) ?? ]
( ?? /2) [2?? '
+ ( ?? - 1) ?? '
]
=
3?? + 8
7?? + 15
 or 
?? + {( ?? - 1) /2}?? ?? '
+ ( ?? - 1) /2?? '
=
3?? + 8
7?? + 15
( ?? ) 
we have to find 
?? 12
?? 12
=
?? +11?? ?? '
+11?? '
 
choosing ( ?? - 1) /2 = 11 or ?? = 23 in (1), 
we get 
?? 12
 T
12
 
'
=
?? +11?? ?? '
+11?? '
=
3( 23) +8
( 23) ×7+15
=
77
176
=
7
16
 
Problem 6: If sum of ?? terms of a sequence is given by ?? ?? = 3?? 2
- 4?? , find its 50
th 
 term. 
Solution: Let ?? ?? is ?? th 
 term of the sequence so ?? ?? = ?? ?? - ?? ?? -1
. 
 = 3?? 2
- 4?? - 3( ?? - 1)
2
+ 4( ?? - 1) = 6?? - 7
 so  ?? 50
= 293
 
Remarks: 
(i) The first term and common difference can be zero, positive or negative (or any complex 
number.) 
(ii) If ?? , ?? , ?? are in A.P. ? 2?? = ?? + ?? & if ?? , ?? , ?? , ?? are in A.P. ? ?? + ?? = ?? + ?? . 
(iii) Three numbers in A.P. can be taken as a - d, a, a + d; four numbers in A.P. can be 
taken as a -3?? , ?? - ?? , ?? + ?? , ?? + 3?? ; five numbers in A.P. are ?? - 2?? , ?? - ?? , ?? , ?? + ?? , ?? + 2?? ; 
six terms in A.P. are ?? - 5?? , ?? - 3?? , ?? - ?? , ?? + ?? , ?? + 3?? , ?? + 5?? etc. 
(iv) The sum of the terms of an A.P. equidistant from the beginning & end is constant and 
equal to the sum of first & last terms. 
(v) Any term of an A.P. (except the first) is equal to half the sum of terms which are 
equidistant from it. ?? ?? = 1/2( ?? ?? -?? + ?? ?? +?? ) , ?? < ?? . For ?? = 1, ?? ?? = ( 1/2) ( ?? ?? -1
+ ?? ?? +1
) ; 
For k = 2, ?? ?? = ( 1/2) ( ?? ?? -2
+ ?? ?? +2
) and so on. 
(vi) If each term of an A.P. is increased, decreased, multiplied or divided by the same non-
zero number, then the resulting sequence is also an AP. 
(vii) The sum and difference of two AP's is an AP. 
Problem 7: The numbers t( t
2
+ 1) , -
t
2
2
 and 6 are three consecutive terms of an A.P. If t be 
real, then find the the next two term of A.P. 
Solution:  2?? = ?? + ?? ? - ?? 2
= ?? 3
+ ?? + 6 
or ?? 3
+ ?? 2
+ ?? + 6 = 0 
or  ( ?? + 2) ( ?? 2
- ?? + 3) = 0 ? ?? 2
- ?? + 3 ? 0 ? ?? = -2 
the given numbers are -10, -2,6 
which are in an A.P. with ?? = 8. The next two numbers are 14,22 
Problem 8: If ?? 1
, ?? 2
, ?? 3
, ?? 4
, ?? 5
 are in A.P. with common difference ? 0, then find the value 
of ?
?? =1
5
??? ?? , when ?? 3
= 2. 
Solution:   As ?? 1
, ?? 2
, ?? 3
, ?? 4
, ?? 5
 are in A.P., we have ?? 1
+ ?? 5
= ?? 2
+ ?? 4
= 2?? 3
. Hence ?
?? =1
5
??? ?? =
10. 
Problem 9: If ?? ( ?? + ?? ) , ?? ( ?? + ?? ) , ?? ( ?? + ?? ) are in A.P., prove that 
1
?? ,
1
?? ,
1
?? are also in A.P. 
Solution:  ? ?? ( ?? + ?? ) , ?? ( ?? + ?? ) , ?? ( ?? + ?? ) are in A.P. ?  subtract ???? + ???? + ???? from each 
-???? , -???? , -???? are in A.P. divide by -?????? 
1
a
,
1
 b
,
1
c
 are in A.P.  
Problem 10: If 
?? +?? 1-????
, ?? ,
?? +?? 1-????
 are in A.P. then prove that 
1
?? , ?? 1
?? are in A.P. 
Solution:  ? 
?? +?? 1-????
, ?? ,
?? +?? 1-????
 are in A.P. 
?? -
?? + ?? 1 - ????
=
?? + ?? 1 - ????
- ?? 
-?? ( ?? 2
+ 1)
1 - ????
=
?? ( 1 + ?? 2
)
1 - ????
 
? - a + abc= c - abc 
?? + ?? = 2?????? 
divide by ac 
1
?? +
1
?? = 2?? ? 
1
?? , ?? ,
1
?? are in A.P.  
Arithmetic mean (mean or average) 
(A.M.): 
If three terms are in A.P. then the middle term is called the A.M. between the other two, so 
if ?? , ?? , ?? are in A.P., ?? is A.M. of ?? & ?? . 
A.M. for any ?? numbers ?? 1
, ?? 2
, … , ?? ?? is; ?? =
?? 1
+?? 2
+?? 3
+?..+?? ?? ?? . 
?? -Arithmetic means between two 
numbers: 
If ?? , ?? are any two given numbers & ?? , ?? 1
, ?? 2
, … , ?? ?? , ?? are in A.P., then ?? 1
, ?? 2
, … ?? ?? are the n 
A.M.'s between ?? & ?? . 
?? 1
= ?? +
?? - ?? ?? + 1
, ?? 2
= ?? +
2( ?? - ?? )
?? + 1
, … … , ?? ?? = ?? +
?? ( ?? - ?? )
?? + 1
 
Note: Sum of ?? A.M.'s inserted between ?? &?? is equal to ?? times the single A.M. between 
?? &?? 
i.e. ?
?? =1
?? ??? ?? = ???? , where ?? is the single A.M. between ?? &?? 
i.e. ?? =
?? +?? 2
 
Problem 11: If ?? , ?? , ?? , ?? , ?? , ?? are A. M's between 2 and 12, then find ?? + ?? + ?? + ?? + ?? + ?? . 
Solution:   Sum of A.M.  
?? = 6 single A.M. =
6( 2+12)
2
= 42 
Problem 12: Insert 10 A.M. between 3 and 80. 
Solution:   Here 3 is the first term and 80 is the 12
th 
 term of A.P. so 80 = 3 + ( 11) d 
? ?? = 7