ON SEQUENCES OF NUMBERS IN GENERALIZED ARITHMETIC AND GEOMETRIC PROGRESSIONS

Transcription

1 Palestine Journal of Matheatics Vol 4) 05), Palestine Polytechnic University-PPU 05 ON SEQUENCES OF NUMBERS IN GENERALIZED ARITHMETIC AND GEOMETRIC PROGRESSIONS Julius Fergy T Rabago Counicated by Ayan Badawi MSC 00 Classifications: Priary B5, B83; Secondary Y55 Keywords and phrases: Sequence of nubers with alternate coon differences, sequence of nubers with alternate coon ratios, the general ter and the su of a sequence of nubers Abstract The paper provides a further generalization of the sequences of nubers in generalized arithetic and geoetric progressions [] Introduction The usual arithetic sequence of nubers takes the for: a, a d, a d, a 3d,, a n )d, a nd, while the geoetric sequence of nubers has the for a, ar, ar, ar 3,, ar n, ar n, Forally speaking, an arithetic sequence is a nuber sequence in which every ter except the first is obtained by adding a fixed nuber, called the coon difference, to the preceeding ter and a geoetric sequence is a nuber sequence in which every ter except the first is obtained by ultiplying the previous ter by a constant, called the coon ratio The sequence, 3, 5, 7, 9,, is an exaple of arithetic sequence with coon difference and the sequence, 4, 8, 6, is a geoetric sequence with coon ratio Certain generalizations of arithetic and geoetric sequence were presenten [], [3], [4] Particularly, in [3], Zhang and Zhang introduced the concept of sequences of nubers in arithetic progression with alternate coon differences ann [4], Zhang, etal provided a generalization of the sequence It was then extended by Majudar [] to sequences of nubers in geoetric progression with alternate coon ratios and the periodic sequence with two coon ratios The author [] also provided a sipler and shorter fors and proofs of soe cases of the results presented by Zhang and Zhang in [3] Recently, Rabago [] further generalized these concepts by introducing additional coon differences and coon ratios Here we will provide another generalization of the sequences of nubers definen [] and [3] by providing a definition to what we call sequences of nubers with alternate coon differences Section ) and sequence of nubers with alternate coon ratios Section 3) Throughout in the paper we denote the greatest integer containen x as x Sequence of nubers with alternate coon differences We start-off this section with the definition of what we call sequence of nubers with alternate coon differences Definition A sequence of nubers {a n } is called a sequence of nubers with alternate coon differences if for a fixed natural nuber and for all j =,,,, a k )j a k )j = d j, for all k N Here d j is the j-th coon difference of {a n } With the above definition, a sequence of nubers with alternate coon differences takes the following for: a, a d, a d d,, a d d d, a d d d, a d d d, a d d d, )

2 ON SEQUENCES OF NUMBERS 7 The sequence, 3, 5, 8, 9,, 4, 5, 7, 0, is an exaple of a sequence of nubers with 3 alternate coon differences The coon differences are d =, d =, and d 3 = 3 Theore Let {a n } be a sequence of nuber that takes the for ) Then, the forula for the n th ter of the sequence {a n } is given by n ) i a n = a ) Proof Obviously, ) holds for n We only need to show that ) is true for n > to prove the validity of the foula Suppose ) holds for soe natural nuber k Hence, k ) i a k = a Let k = p ) j and p N Now, for every j =,,, N, we have a k = a k d j Thus, k ) i a k = a d j = a p ) j ) i d j j = a p j i j = a p p ) d j = a = a = a = a j p j p ) p j i p j i d j p j i j p ) j) ) i p ) j) ) i k ) ) i Below is a table of forulas for the n th ter a n of the given sequence for specific values of n th ter a n a n )d n n a d d n n n 3 a d d d n n n n 4 a d d d 3 d n 3 n n n n 5 a d d d 3 d 4 d

3 7 Julius Fergy T Rabago Corollary 3 Let and n be natural nubers If n ) then we have ) n a n = a Proof Suppose n ) then n = k for soe k N Then, a n = a k i = a k ) n = a Corollary 4 If n, we have n a n = a ) d Proof Suppose n then n = k for soe k N So, a n = a k i = a k n d = a ) d Lea 5 For any natural nubers and n, we have Proof Note that Hence, n ) i i n = n = n ) i n = k k n < k = n ) n n = k k k k ) = k = n Theore 6 If = d for i, we have ) n a n = a n d Proof Let = d, for i, in ) Hence, a n = a n ) i n n ) i n = a d d n ) n 3) n ) = a d { ) } n ) i n n = a d n n = a n ) d d d 3) n d d

4 ON SEQUENCES OF NUMBERS 73 Theore 7 Let {a n } be a sequence of nuber that takes the for ) Then, the forula for the su of the first n ters of the sequence {a n } is given by where N = n j= j= S n = na N i N i ) N i, 4) Proof Consider a sequence {a n } that takes the for ) Then, ) j ) i j ) i a j = a = na = na n ) i Letting N = n, conclusion follows n i j= n i) ) Theore 8 The su of the first n ters of the sequence {a n } that takes the for ) with = d, for i, is given by nn ) n S n = na d d d ) n ) n 5) Proof Consider a sequence {a n } that takes the for ) Then, a j = j= j= = na = na = na a j nn ) d nn ) j j= ) d j d ) j d j j= d j d d d ) j= n n n nn ) d d d ) ) 3 Sequence of nubers with alternate coon ratios We define the sequence of nubers with alternate coon ratios {a n } as follows: Definition 3 A sequence of nubers {a n } is called a sequence of nubers with alternate coon ratios if for a fixed natural nuber and for all j =,,,, a k )j a k )j = r j, for all k N Here r j is the j-th coon ratio of {a n } With the above definition, we can see iediately that a sequence of nubers {a n } with alternate coon ratios has the following for: a, ar, ar r,, ar r r, ar r r, ar r r, ar r r, 3) The sequence,, 6, 4, 48, 44, 576, 5, is an exaple of a sequence of nubers {a n } with 3 alternate coon ratios The coon ratios are r =, r = 3, and r 3 = 4

5 74 Julius Fergy T Rabago Theore 3 Let {a n } be a sequence of nuber that takes the for 3) Then, the forula for the n th ter of the sequence {a n } is given by where e i = n ) i a n = a r ei i, 3) In particular, if n ), we have anf n ), a n = a a n = a r r i ) n ), 33) ) ) n r i 34) Proof The proof is by induction on n Obviously, 3) holds for n We will show that 3) is true for n > Suppose 3) holds when for soe natural nuber k That is, where e i = p )j ) i a k = a Let k = p )j and p N Now, for every j =,,, N, we have a k = a k r j Thus, p ) j ) i a k = a r ei i r j, where e i = j = a r ei i j = a r p i = a j = a r fi i r ei i, r ei i r j, where e i = p j i r p i r j r hi i, where h i = r gi i, where f i = p j i, g i = p j i k ) ) i If n resp n )) then 33) resp 34)) follows iediately Theore 33 Consider a sequence {a n } that takes the for 3) and suppose r i = r, for i Then, a n = a r e r, s 35) where e = n n and s = n Proof Let r = r i, for i, in 3) Hence, n ) i a n = a r ei i, where e i = = a r e r s, where e = n ) i But, by Lea 5), e = n n Thus, an = a r e rs and s = n

6 ON SEQUENCES OF NUMBERS 75 Theore 34 Let {a n } be a sequence of nuber that takes the for 3) Then, the forula for the su of the first n ters of the sequence {a n } is given by r e n )) ) i S n = a R a r e n N i N i j r k, r where R = i j= r j, r = r i, e n = n and N = n j= k= Proof Consider a sequence {a n } that takes the for 3) and let R = i j= r j, r = r i, p = e n = n then n a j = a r ej i where e j = j ) i j= Expanding the expression, we obtain j= j= p a j = a a R r j a r e n 3) r e nn 4) r e n a r e n ) r e n ) r e n j=0 Siplifying and rewriting the expression in copact for, we obtain )) r p i a j = a R a r p M i r j= j j= k= where M i = N i N i ) which is the desired result Theore 35 Let {a n } be a sequence of nuber that takes the for 3) with r i = r, for all i Then, the forula for the su of the first n ters of the sequence {a n } is given by ) ) ) r S n = a r r ) p ) p r n p r r a r r, r r where p = n Proof Consider a sequence {a n } that takes the for 3) with r i = r, for all i and let p = n, n a j = a j= which is desired j= j= r j r r = a r j r r = a j= ) j r r ) p p r j ) r p r r ) { r = a r ) j= r j j=p ) r j ) r r r r j j= ) r p n p a r r j= j= r r a r r j a r j ) ) r = a r r ) p r r r r r r r r r ) 3 j= ) p n j=p ) r r r r ) p n p j= j= r j r k r j r j r j ) r ) } r r p r ) r a r r ) p r ) r n p, r

7 76 Julius Fergy T Rabago 4 Soe Rearks If we replace by t in ) and define t as the period of the sequence {a n } and by considering = d for i as the first coon difference of the sequence and d = d as the second difference then we obtain, ) n n a n = a n d d t t 4) Equation 4) is exactly the forula for the n th ter of a periodic nuber sequence with two coon differences obtained by Zhang and Zhang in [4] Furtherore, it can be observed fro 4) that a n a n )d as Siilarly, if = d for all i, a n = a n )d In 5), on the other hand, would have S n = na nn ) d if and a siilar result will be obtainef = d for all i Also, note that in 3), a n a r n if we apply the sae arguent letting either or r i = r for all i Furtherore, the liit of the su given by References j= a r n r r ) n a r n r ) as [] AAK Majudar, Sequences of nubers in generalized arithetic and geoetric progressions, Scientia Magna, 4 008), No, 0- [] JFT Rabago, Sequence of nubers with three alternate coon differences and coon ratios, Int J of Appl Math Res, 0), No3, [3] X Zhang and Y Zhang, Sequence of nubers with alternate coon differences, Scientia Magna, 3 007), No, [4] X Zhang, Y Zhang, and J Ding, The generalization of sequence of nubers with alternate coon differences, Scientia Magna, 4 008), No, 8- Author inforation Julius Fergy T Rabago, Departent of Matheatics and Coputer Science, College of Science, University of the Philippines, Baguio Governor Pack Road, Baguio City 600, PHILIPPINES E-ail: jfrabago@gailco Received: Deceber 3, 03 Accepted: April 7, 04