1 / 15

Counting Add Summation

Counting Add Summation. An Introduction. 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: Sequences Series . Sequence.

gabi
Download Presentation

Counting Add Summation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Counting Add Summation An Introduction

  2. 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: • Sequences • Series

  3. 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”.

  4. 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

  5. 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.

  6. Series • 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+………... }Finite Series }Infinite Series

  7. 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

  8. 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 ,……. nthterms respectively, then we have. T2 – T1 = d T3 – T2 = d T4 – T3 = d ……………. ……………. Tn – Tn-1 = d • General Term= Tn = a+(n-1)d

  9. General terms of A.P. • If l is the last term of A.P., then the number of terms in the A.P. is • n= l-a+d /d • And the common difference d= l-a/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.) l-a d=--------- n-1 l-a+d n= -------- d

  10. 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 : • 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}

  11. 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}

  12. 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.

  13. General terms of G.P. • Let a be the first term and r bethe common ratio of a G.P. let T1 , T2 , T3 ,…………. Tn denote 1st , 2nd ,3rd ,……. nthterms respectively, then we have. T2 = T1r T3= T2r T4 =T3r ……………. ……………. Tn=Tn-1r • General Term= Tn = arn-1

  14. 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: • If r <1 • If r ≤ -1 or r >1 • If r=1

  15. Summation

More Related