Sequences
and Series
Sequence
Sequence is a function whose domain is the set of natural numbers or
some subset of the type { 1, 2, 3,... , k}. We represents the images of
1, 2, 3,K , n , ... as f1 , f2 , f3 ,... , fn ... , where fn = f ( n ).
In other words, a sequence is an arrangement of numbers in definite
order according to some rule.
l A sequence containing a finite number of terms is called a finite
sequence.
l A sequence containing an infinite number of terms is called an
infinite sequence.
l A sequence whose range is a subset of real number R, is called a
real sequence.
Progression
A sequence whose terms follow a certain pattern is called a progression.
Series
If a1 , a2 , a3 ,... , an ,... is a sequence, then the sum expressed as
a1 + a2 + a3 + ... + an +... is called a series.
l A series having finite number of terms is called finite series.
l A series having infinite number of terms is called infinite series.
Arithmetic Progression (AP)
A sequence in which terms increase or decrease regularly by a fixed
number. This fixed number is called the common difference of AP.
e.g. a, a + d, a + 2d,... is an AP, where a = first term and d = common
difference.
, nth Term (or General Term) of an AP
If a is the first term, d is the common difference and l is the last term
of an AP, i.e. the given AP is a , a + d , a + 2d , a + 3d,..., l, then
(a) nth term is given by an = a + ( n − 1)d
(b) nth term of an AP from the last term is given by an′ = l − ( n − 1)d
Note
(i) an + an′ = a + l
i.e. nth term from the begining + nth term from the end
= first term + last term
(ii) Common difference of an AP
d = an − an − 1, ∀ n > 1
1
(iii) an = [an − k + an + k ], k < n
2
Properties of Arithmetic Progression
(i) If a constant is added or subtracted from each term of an AP, then
the resulting sequence is also an AP with same common
difference.
(ii) If each term of an AP is multiplied or divided by a non-zero
constant k, then the resulting sequence is also an AP, with
d
common difference kd or respectively, where d = common
k
difference of given AP.
(iii) If an , an + 1 and an + 2 are three consecutive terms of an AP, then
2an + 1 = an + an + 2.
(iv) If the terms of an AP are chosen at regular intervals, then they
form an AP.
(v) If a sequence is an AP, then its nth term is a linear expression in
n, i.e. its nth term is given by An + B, where A and B are
constants and A = common difference.
Selection of Terms in an AP
(i) Any three terms in AP can be taken as
( a − d ), a , ( a + d )
(ii) Any four terms in AP can be taken as
( a − 3d ), ( a − d ), ( a + d ), ( a + 3d )
(iii) Any five terms in AP can be taken as
( a − 2d ), ( a − d ), a , ( a + d ), ( a + 2d )
and Series
Sequence
Sequence is a function whose domain is the set of natural numbers or
some subset of the type { 1, 2, 3,... , k}. We represents the images of
1, 2, 3,K , n , ... as f1 , f2 , f3 ,... , fn ... , where fn = f ( n ).
In other words, a sequence is an arrangement of numbers in definite
order according to some rule.
l A sequence containing a finite number of terms is called a finite
sequence.
l A sequence containing an infinite number of terms is called an
infinite sequence.
l A sequence whose range is a subset of real number R, is called a
real sequence.
Progression
A sequence whose terms follow a certain pattern is called a progression.
Series
If a1 , a2 , a3 ,... , an ,... is a sequence, then the sum expressed as
a1 + a2 + a3 + ... + an +... is called a series.
l A series having finite number of terms is called finite series.
l A series having infinite number of terms is called infinite series.
Arithmetic Progression (AP)
A sequence in which terms increase or decrease regularly by a fixed
number. This fixed number is called the common difference of AP.
e.g. a, a + d, a + 2d,... is an AP, where a = first term and d = common
difference.
, nth Term (or General Term) of an AP
If a is the first term, d is the common difference and l is the last term
of an AP, i.e. the given AP is a , a + d , a + 2d , a + 3d,..., l, then
(a) nth term is given by an = a + ( n − 1)d
(b) nth term of an AP from the last term is given by an′ = l − ( n − 1)d
Note
(i) an + an′ = a + l
i.e. nth term from the begining + nth term from the end
= first term + last term
(ii) Common difference of an AP
d = an − an − 1, ∀ n > 1
1
(iii) an = [an − k + an + k ], k < n
2
Properties of Arithmetic Progression
(i) If a constant is added or subtracted from each term of an AP, then
the resulting sequence is also an AP with same common
difference.
(ii) If each term of an AP is multiplied or divided by a non-zero
constant k, then the resulting sequence is also an AP, with
d
common difference kd or respectively, where d = common
k
difference of given AP.
(iii) If an , an + 1 and an + 2 are three consecutive terms of an AP, then
2an + 1 = an + an + 2.
(iv) If the terms of an AP are chosen at regular intervals, then they
form an AP.
(v) If a sequence is an AP, then its nth term is a linear expression in
n, i.e. its nth term is given by An + B, where A and B are
constants and A = common difference.
Selection of Terms in an AP
(i) Any three terms in AP can be taken as
( a − d ), a , ( a + d )
(ii) Any four terms in AP can be taken as
( a − 3d ), ( a − d ), ( a + d ), ( a + 3d )
(iii) Any five terms in AP can be taken as
( a − 2d ), ( a − d ), a , ( a + d ), ( a + 2d )
