1.Algebra Booster

Sequence and Series 1.1
CHAPTER
1 Sequence and Series
1. INTRODUCTION
CONCEPT BOOSTER
It is formally defined as, a sequence is a set of numbers arranged
in a definite order so that all the numbers after the first
will maintain a definite rule.
For example, 1, 3, 5, ... is a list of odd natural numbers. It
can be written as 2n –1, where n Œ N.
So a sequence may be defined like a function by a variable
expression.
Sequence
It is a function whose domain
is the set of natural numbers.
It can also be defined as a
function whose range is the
set of real numbers.
Let f(n) = n + 2, where n ΠN
Thus, {3, 4, 5, 6, 7, …} is a
sequence.
1.1 Series
If a sequence is connected by either +ve sign or –ve sign or
both, it is called a series.
For example, 1 + 2 + 3 + 4 + … is a series.
1.2 Finite and Infinite Sequences
A sequence is said to be finite or infinite according as the
number of elements is finite or infinite.
For example, {1, 3, 5, 7, …, 99} is a finite sequence.
2. PROGRESSION
It is a sequence in which each term is followed by a certain
pattern.
There are four types of progression, namely, AP, GP, HP
and AGP.
N
n
f
R
fn ()
2.1 Arithmetic Progression (AP)
If the algebraic difference between any two consecutive
terms is same throughout the sequence, then it is called an
A.P sequence.
i.e. t n+1
– t n
= constant = d, which is known as the common
difference.
2.2 nth Term from the Beginning of an AP Sequence
It is given as t n
= a + (n – 1)d,
where a is the first term and d the common difference of the
AP.
Which is also known as the general term of A.P
2.3 nth Term from the end of an AP Sequence
Let the first term of a sequence is a, common difference d and
the number of terms is m of an AP.
Thus, T n
= l – nd = a + (m – n)d,
where l is the last term
2.4 nth Term of an AP Sequence is a Linear
Expression of n.
Let the first term is a and the common difference is d of an AP.
Then t n
= a + (n – 1)d
= (a – d) + dn = A + Bn
where A = a – d and B = d.
which is a linear expression of n.
2.5 Sum of n-terms of an AP Sequence
Let first term is a and the common difference is d of an AP.
n
\ The sum, Sn
= [2 a + ( n-1) d]
2
2.6 Properties of AP Sequences
Property I
Let a 1
, a 2
, a 3
, …, a n
be in AP and k be a non-zero real number,
then
(i) a 1
+ k, a 2
+ k, a 3
+ k, …, a n
+ k are also in AP.