![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](https://image.slidesharecdn.com/vishwaadarshacolageitahari2sunsari-210224131031/85/sequence-and-series-presentatation-12-320.jpg)
sequence and series presentatation
- 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 ?