this is the presentation about math's topic sequence and series

1. VISHWA ADARSHA COLAGE
ITAHARI 2 SUNSARI
PRESENTED BY :
PAWAN DHAKAL
SUBMITED TO :
MR. SITARAM SUBEDI
PRESENTATION OF MATHEMATICS ON SEQUENCE AND SERIES
BCA FIRST SEMESTER
2ND BATCH

2. CONTENTS
 INTRODUCTION OF SEQUENCE AND SERIES
 TYPES OF SEQUENCE AND SERIES
 INTRODUCTION OF HARMONIC PROGERESSION AND
HARMONIC MEAN
 QUESTION AND SOLUTION OF HARMONIC
PROGRESSION
 HARMONIC MEAN AND IT’S FORMULA
 USE OF HARMONIC PROGRESSION

3.  INTRODUCTION OF SEQUENCE AND SERIES
 SEQUENCE:It is the order of terms kept or
arranged by a specific rule or condition. For
example
i. 1, 4, 7, 10,……
ii. 2, 4, 8, 16, ….
There are two type of sequence :
1. Finite sequence : The sequence which has a last
term is finite sequence. Like a1,a2,a3,a4,….,an
2. Infinite sequence : The sequence which doesn't
have a last term is called infinite sequence. Like
a1,a2,a3,a4,……

4.  SERIES
Series It is the sequence of terms connected by the + or –
sign is called series. For example : 1+2+3+4+5+…. And There
are two type of series :
1. Finite series : A finite sequence whose terms are connected
by + or – sign is called finite series. Like a1+a2+a3+a4+….+an
2. Infinite series : An infinite sequence whose terms are
connected by + or – sign is called infinite series. Like
a1+a2+a3+a4….

5.  TYPES OF SEQUENCES/ PROGESSION
 ARTHMETIC PROGRESSION(AP):If a sequence or series terms are
increasing or decreasing by a constant difference then it is known as
arithmetic progression. For example : 1,2,3,4,5,6,….. and 1+2+3+4+5+6+….
 Geometric progression : If a sequence or series terms having first term
non-zero and the ratio of any term to its preceding term bears a constant.
For example : 2,4,8,16,32…. and 3+9+27+81…..
 Harmonic progression : The reciprocal of arithmetic progression is
called harmonic progression. For example :1/2,1/3,1/4,1/5,…. form a H.P. as
the reciprocal of its terms 2,3,4,5,…. forms a A.P. with difference 1.

6.  DETAILS ABOUT HARMONIC SEQUENCE /PROGRESSION
 A SEQUENCE OR A NUMBER IS SAID
TO BE IN HARMONIC SEQUENCE OR
PROGRESSION IF THE RECIPROCAL
OF IT’S TERM FORMS AN A.S.
 EXAMPLES OF ARTHMETIC
SEQUENCE/PROGRESSION:
i. 1,
1
2
,
1
3
,
1
4
, … … . .FORM A H.P . AS A
RECIPROCAL OF IT’S TERMS 1, 2, 3,
4,… FORM AN A.S. WITH COMMON
DIFFERENCE (d)=1

7.  PROPERTIES OF HARMONIC SEQUENCE:
IF EACH TERM OF H.S. BE MULTIPLIED OR DIVIDES BY A CONSTANT
NUMBER, THE SEQUENCE OF THE RESULTING NUMBER IS ALSO IN H.P.
THAT IS ,a,b,c,d……BE IN H.S AND K IS ANY NON ZERO CONSTANT, THEN
1. ak, bk, ck, dk,…….are on h.S.
2.
𝑎
𝑘
,
𝑏
𝑘
,
𝑐
𝑘
,
𝑑
𝑘
, … … …ARE ALSO IN H.S.

8.  HARMONIC MEAN
• IF THREE NUMBER A,H,B ARE IN HARMONIC SEQUENCE , THE H
IS SAID TO BE THE HARMONIC MEAN BETWEEN A AND B.

9. OR,
2
𝐻
=
𝑎+𝑏
𝑎𝑏
THEREFORE 𝐻 =
2𝑎𝑏
𝑎+𝑏
 HOW TO FIND THE HARMONIC MEAN BETWEEN TWO GIVEN
NUMBERS
LET a AND b BE THE TWO NUMBER AND H BE THE HARMONIC MEAN
BETWEEN THEM. HENCE
1
𝑎
,
1
𝐻
,
1
𝑏
FORM AN AP.
THIS IMPLIES, 1
𝐻
−
1
𝑎
=
1
𝑏
−
1
𝐻
OR ,
2
𝐻
=
1
𝑎
+
1
𝑏

10. Formulas for H.M
Let a and b be two given numbers and H the harmonic mean between
them. Then let a, H, b form a H.P and 1/a, 1/H, 1/b form an A.P
Then, from the definition of A.P
t2-t1=t3-t2
1/H-1/a= 1/b-1/H
2/H= 1/a+1/b
2/H= a+b/ab
H=2ab/a+b

11. RELATION BETWEEN AM , GM AND HM .
The A,M, G.M and H.m between any two unequal positive numbers satisfy the
following relations:
a) (G.M)2=A.M * H.M
b) A.M > G.M> H.M
Let a and b be any 2 unequal positive numbers
we know
A.M=a+b/2, G.M=√ab, H.M=2ab/a+b
3) To prove,(G.M)2= A.M *H.M
R.H.S = A.M * H.M
=a+b/2 * 2ab/a+b
=ab
=(√ab)2
We know, G.M=√ab
And (G.M)2=(√ab)2
Hence (G.M)2= A.M * G.M proved.

12. 4) To prove, A.M>G.M>H.M
Here,
A.M-G.M=a+b/2 - √ab
=(a+b - 2√ab)/2
=[(√a)]2 - 2√ab +(√b)2]/2
=(√a - √b)2/2 which is greater than 0
since, A.M – G.M>0
so A.M> G.M
Again, G.M – H.M= √ab – 2ab/a+b
= √ab (1- 2ab/a+b)
= √ab (a+b – 2√ab)/a+b
= √ab (√a - √b)2/a+b which is greater than o
since, G.M – H.M>0
so G.M > H.M
Now combining the results we get,
A.M > G.M > H.M

13. USE OF HARMONIC MEAN
i. Harmonic means are used in finance to average data like
price multiples.
ii. Harmonic means can also be used by market technicians
to identify patterns such as Fibonacci sequences.
iii. Helps to find out average speed of a journey i.e,
iv. average speed when the distance is covered= HM of the
speed
v. Leaning the tower of lire in which the blocks are stacked
1/2, 1/4, 1/6,…distance sideways below the original block.
vi. Used in geometry.

14. ANY QUESTION ?