COUNTING
 Here
counting doesn’t mean
counting the things
physically.
 Here we will learn, how we
can count without counting.
 Two basic things regarding
counting are:
1.
2.
Sequences
Series
SEQUENCE


Let N be the set of natural
numbers and Nn be the set
of first n natural numbers,
i.e.,
Nn ={1,2,3,4,……….n}
and X be a non-empty set,
then a map f:Nn X is
called a finite sequence and
a map f:N X is called an
infinite sequence.
A sequence following some
definite rule (or rules) is
called a “progression”.
SEQUENCE

Illustrations:
}
}
3,5,7,9,……………….,21
 8,5,2,-1,-4,………..,16
 2,6,18,54,………,1458

1,4,9,16,………………
 1,3,5,7,9,……………..
 8,5,2,-1,-4,………..

Finite sequence
Infinite sequence
SERIES
If the terms of
sequence are
connected by plus (or
minus) signs, a series
is formed.
 Thus, if Tn denotes the
general term of a
sequence, then T1 + T2
+ T3 + T4 +………+ Tn
is a series of n terms.

SERIES

Illustrations:
}
}
3+5+7+9+…………+21
 8+5+2+1+4+………+16
 2+6+18+54+………+14
58

1+4+9+16+………….
 1+3+5+7+9+………...

Finite Series
Infinite Series
ARITHMETIC PROGRESSION
Arithmetic Progression(A.P.): A sequence (finite
or infinite) is called an arithmetic progression (
abbreviated A.P.). Iff the difference of any term from
its preceding term is constant.
 This constant is usually denoted by d and is called
common difference.
 The first term of A.P. is usually denoted by a.
 for example: The sequence 3,5,7,9,…..21 is a finite A.P.
with d=2
 and the sequence 8,5,2,-1,…….. Is an infinite A.P. with
d= -3

GENERAL TERMS OF A.P.
Let a be the first term and d the common difference
of an A.P. let T1 , T2 , T3 ,…………. Tn denote 1st , 2nd
,3rd ,……. nth terms respectively, then we have.
T2 – T1 = d
T3 – T2 = d
T4 – T3 = d
…………….
…………….
Tn – Tn-1 = d
 General Term= Tn = a+(n-1)d

GENERAL TERMS OF A.P.

If l is the last term of A.P., then the number of terms
in the A.P. is
l-a+d
n= l-a+d /d
 And the common difference
d= l-a/n-1


n= -------d
l-a
d= --------n-1
If a1 , a2 , a3 ,……….. ,an are non-zero numbers such that 1/
a1 ,1/ a2 ,1/ a3 ,…………, 1/ an are in A.P., then a1 , a2 , a3
,……… ,an are said to be in “Harmonic Progression”
(abbreviated H.P.)
SUM OF N TERMS OF AN A.P.

let a be the first term,
d the common
difference and l the
last term of the given
A.P. then Sn , sum of n
terms of this A.P. can
be calculated by three
different mechanism
depending upon input
values :
1.
2.
3.
If a, n and d is given
If a, n and l is given
If n, l and d is given
Sn = n/2{2l-(n-1)d}
ARITHMETIC MEANS A.M.
•When three numbers
are in A.P., the middle
one is said to be the
Arithmetic Mean
between the other two.
• if a,b and c are in A.P.,
then A.M.= b
•To find the A.M.
between two given
numbers A= a+b/2 where
a and b are two given
numbers.
• if we have n terms
between a and b then
A= n{(a+b)/2}
GEOMETRIC PROGRESSION G.P.

A sequence (finite or
infinite) of non-zero
terms is called a
“geometric
progression” iff the
ratio of any terms to
its preceding term is
constant.
this (non-zero) constant is usually denoted by r
and is called “Common ration”
 We assume that none of the terms of the
sequence is zero.
 Eg. a,ar,ar2 , ar3 , ar4 ,…………., arn-1 is in
G.P.

GENERAL TERMS OF G.P.
Let a be the first term and r be the common ratio of a
G.P. let T1 , T2 , T3 ,…………. Tn denote 1st , 2nd ,3rd
,……. nth terms respectively, then we have.
T2 = T1r
T3= T2r
T4 =T3r
…………….
…………….
Tn=Tn-1 r
 General Term= Tn = ar n-1

SUM OF N TERMS OF A G.P.

Sum of n terms of a
G.P. depends upon the
value of r. there are
three possibilities as:
1.
2.
3.
If r <1
If r ≤ -1 or r >1
If r=1
SUMMATION