IDENTITY OF AN UNKNOWN TERM IN A TETRANACCI- LIKE SEQUENCE

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IDENTITY OF AN UNKNOWN TERM IN A TETRANACCI- LIKE SEQUENCE Gautam S Hathiwala 1, Devbhadra V Shah 2 Lecturer, Department of Applied Science, CK Pithawalla College of Engineering & Technology, Surat,

1 IDENTITY OF AN UNKNOWN TERM IN A TETRANACCI- LIKE SEQUENCE Gautam S Hathiwala 1, Devbhadra V Shah 2 Lecturer, Department of Applied Science, CK Pithawalla College of Engineering & Technology, Surat, Gujarat India 1 Associate Professor, Head of the Department of Mathematics, Sir PT Sarvajanik College of Science, Surat, Gujarat, India 2 Abstract: Given any four arbitrary real numbers and, we insert terms between and such that these terms satisfies the tetranacci recurrence relation We derive the general formula for all these terms inserted between and Keywords: Fibonacci sequence, Inserting of terms, Row Operations, Tetranacci sequence I INTRODUCTION Many types of sequences have been studied since last few centuries May it be arithmetic, geometric, harmonic or Fibonacci sequence; they have been very well defined Also many interesting results have been obtained related to them Analogous to Fibonacci sequence, sequences like tribonacci, tetranacci and pentanacci have been defined and their properties have been studied Fibonacci sequence 1,1,2,3,5,8,13,21, is a succession of terms obtained by adding the preceding two terms A derivative of this sequence can be called Fibonacci-like sequence The only difference is that it can start with any two arbitrary terms, whereas the successive terms are obtained by adding previous two terms Similarly we can consider a tetranacci sequence defined by the recurrence relation ; provided and First few terms of this sequence are 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, Much like Fibonacci sequence, we can have derivative for tetranacci sequence If we consider 2,4,1,7 as first four terms of tetranacci like sequence, then using similar recurrence relation, we get a sequence of terms as 2, 4, 1, 7, 14, 26, 48, 95, 183, 352, 678, Generally Fibonacci-like sequence can be expressed as ; where and are any arbitrary real numbers In a like manner we can also define tetranaccilike sequence by the recurrence relation ; and and are arbitrary real numbers In the recent past, much of research have been pursued for the sequence of tribonacci numbers which is obtained by adding previous three terms, provided first three terms are 0, 0, 1 (For further details see [3], [5], [6]) P Howell [4] presented a proof All rights reserved by wwwijaresmnet ISSN :

2 for finding term of the Fibonacci sequence using vectors and eigen-values Also, M Agnes [1] provided a formula for three consecutive terms included in the Fibonacci-like sequence L Natividad [7] considered two arbitrary terms and and inserted terms between a and b such that these terms make a Fibonacci-like sequence Using the Binet s formula, L Natividad [7] obtained the formula for and proved that For example, if we consider and and if, it means that we insert 5 terms between a and b Hence the 7 terms sequence forms Fibonacci-like sequence Then We know that and Placing these values in the identity we get, This helps to get all the successive values Moreover, ACF Bueno [2] went one step ahead and derived an identity to obtain the term of L Natividad s [7] Fibonacci-like sequence More precisely, ACF Bueno [2] considered arbitrary terms and and inserted terms between and to make L Natividad s [7] Fibonacci-like sequence Thus the Fibonacci-like sequence considered becomes Considering this, ACF Bueno [2] derived In this paper, we derive the result for unknown terms inserted between and given arbitrary terms such that the sequence of terms formed satisfy the recurrence relation for the tetranacci-like sequence Given II PRELIMINARIES first four tetranacci numbers, tetranacci sequence of numbers are defined by the recurrence relation We consider four arbitrary terms and insert n terms between such that these terms terms satisfy the tetranacci recurrence relation This gives a sequence of which makes this as tetranacci-like sequence It is interesting to note that the - terms cannot be inserted between or between If it is done so, then by tetranacci recurrence relation, all the gets eliminated For example, if we take and if we insert the between then we have tetranacci-like sequence Hence we get By solving this system of linear equations in two unknowns, the unknown eliminated This is true for all positive integers n Similarly, if for any are inserted between and then they get eliminated This gives the following result Theorem 1: Given four arbitrary terms and if we insert terms where satisfied, then the between or such that the tetranacci recurrence relation is terms inserted always gets eliminated All rights reserved by wwwijaresmnet ISSN : gets

3 In this paper we wish to obtain the identity of the term of the terms inserted between terms of the given terms, such that these terms satisfy the tetranacci recurrence relation In fact we want to prove that for every and (1) where and are arbitrary terms and are the terms inserted between and and are tetranacci numbers Now given four arbitrary terms suppose we insert one term between Hence the sequence of terms is Since this sequence of terms satisfies the tetranacci recurrence relation, we have Solving for we get This can also be verified by considering and in (1): On simplification, we get This verifies (1) for Similarly, (1) can be verified in the same way for where This can be summarized in the form of a theorem as follows: Theorem 2: Given four arbitrary terms and, if we insert terms between and such that the sequence of terms Next we prove that the above result is true for every satisfies tetranacci recurrence relation, then for a fixed ; where III MAIN RESULT Theorem 3: Given four arbitrary terms and we insert terms (where ) between and such that the sequence of terms satisfies tetranacci recurrence relation Then for given, we have where Proof: Here, we are given arbitrary terms and Also, for we insert terms between and such that the sequence of terms satisfies the tetranacci recurrence relation Hence we have system of n linear equations in n unknowns given as All rights reserved by wwwijaresmnet ISSN :

augmented matrix gets transformed to row

4 (2) Writing above system of linear equations in matrix form, we get (3) where ; Hence, the augmented matrix can be written as (4) We prove that the above augmented matrix gets transformed to row echelon form All rights reserved by wwwijaresmnet ISSN :

5 (5) by applying the following row-operations on (4) (1) (i) (ii) (iii) (2) (i) (3) (i) (ii) Perform the operations (i), (ii) and (iii) for and : (i) (4) (ii) (iii) where for (5) Perform operations (i) and (ii) only, for if (6) Perform operation (i) only, for if Table 1 We apply principal of mathematical induction on the row operations given in table 1 to convert system (4) into (5) ie we show that these row operations are sufficient to convert system (4) into row echelon form as given by (5) This will lead to the justification of the identity where for every If we consider then for given arbitrary terms and we insert between and such that the sequence of terms satisfies the tetranacci recurrence relation Thus we have the following system of linear equations: All rights reserved by wwwijaresmnet ISSN :

6 Writing above system in matrix form, we get where ; ; The augmented matrix is (6) Now we convert above system in to the row echelon form Let denote following row operation: row multiplied by real and added to row We apply the row operations given below in table 2 in the given order on (6) (1) (i) (ii) (iii) (2) (i) (3) (i) (ii) (4) (i) (ii) (iii) (5) (i) (ii) (6) (i) Table: 2 Thus (6) gets converted in row echelon form as given below: All rights reserved by wwwijaresmnet ISSN :

where Clearly, the row operations that are performed to convert system (6) into above row echelon form follows those given in table 1 for

by the back substitution, remaining values of can be obtained as and, which Hence we get all values of It can be seen that the obtained

if we insert terms between and then we have a sequence of terms given as and it satisfies tetranacci recurrence relation Thus we have

7 where Clearly, the row operations that are performed to convert system (6) into above row echelon form follows those given in table 1 for Hence, the result is true for Moreover, from (7) by back substitution we get, on simplification gives It can be verified on same line that by the back substitution, remaining values of can be obtained as and, which Hence we get all values of It can be seen that the obtained values of are satisfied by (1) for Now, suppose that the row operations given in table 1 are true for That means given arbitrary terms and if we insert terms between and then we have a sequence of terms given as and it satisfies tetranacci recurrence relation Thus we have system of linear equations given as Writing this system of linear equations in matrix form we get All rights reserved by wwwijaresmnet ISSN :

8 where Hence the corresponding augmented matrix is given as Since, by induction hypothesis, the row operations in table 1 are true for we apply these row operations in the given order on above augmented matrix as given below: (1) (i) (ii) (iii) (2) (i) (3) (i) (ii) All rights reserved by wwwijaresmnet ISSN :

(4) Perform the operations (i), (ii) and (iii) for and : (i) (ii) (iii) (5) Perform operations (i) and (ii) only, for if (6) Perform operation (i) only, for if Table 3 By applying row operations in

9 (4) Perform the operations (i), (ii) and (iii) for and : (i) (ii) (iii) (5) Perform operations (i) and (ii) only, for if (6) Perform operation (i) only, for if Table 3 By applying row operations in table 3 on the above system row echelon form as given below, it gets converted into where, for (8) Consequently, by back substitution from (8), we get all the values of for as given below: Writing all the values of for in compact form, we get for All rights reserved by wwwijaresmnet ISSN :

Now, let Thus, given arbitrary terms and, we

system of linear equations given as Therefore,

where and Writing the corresponding augmented

10 Now, let Thus, given arbitrary terms and, we insert terms between and such that the sequence of terms so formed satisfies tetranacci recurrence relationthus we have system of linear equations given as Therefore, writing above system in matrix form we get where and Writing the corresponding augmented matrix we get (9) All rights reserved by wwwijaresmnet ISSN :

echelon form of augmented system (9) Clearly, it is of the form (5) for operations which we applied on (9) in given order: (1) (i)

11 By induction hypothesis, we can apply the row operations in table 3 on above augmented matrix which gives Also, applying the row operations (i) (ii) (iii) on above augmented matrix in given order, we get (10) where for The above augmented matrix is the row echelon form of augmented system (9) Clearly, it is of the form (5) for operations which we applied on (9) in given order: (1) (i) (ii) (iii) (2) (i) (3) (i) (ii) and it has been obtained by the following row All rights reserved by wwwijaresmnet ISSN :

12 every IJARESM (4) Perform the operations (i), (ii) and (iii) for and : (i) (ii) (iii) (5) Perform operations (i) and (ii) only, for if (6) Perform operation (i) only, for if Table 4 Thus the row operations given in table 1 are also valid for to convert (4) into (5)Moreover, by back substitution from (10), we get and hence for Also, Using the value of for and we get on simplification Similarly, we get by back substitution Writing all these in compact form, we get where Thus, for given arbitrary terms and when we insert terms where between and such that the sequence of terms so formed equations (2), for which the matrix form satisfies tetranacci recurrence relation, we have system of linear is (3) and the corresponding augmented matrix is (4) Then, for every (4) can be converted into row echelon form given in (5) by applying the row operations in table 1 in the same order Thus for every we have All rights reserved by wwwijaresmnet ISSN :

where for Therefore, by back substitution we can write, This gives Also, This gives On simplification we get Similarly we get, by back substitution Writing these values in compact form, we have for

13 where for Therefore, by back substitution we can write, This gives Also, This gives On simplification we get Similarly we get, by back substitution Writing these values in compact form, we have for every where The following result follows immediately from theorem 2 and theorem 3 Theorem 4: Given arbitrary terms and if we insert n terms between and in the given order, such that the tetranacci recurrence relation is satisfied, then for any and we have ACKNOWLEDGEMENT We are grateful to Rinkesh Saliya, the former faculty in the Department of Applied Sciences and Humanities, CK Pithawalla College of Engineering and Technology, Surat for the fruitful discussions during the preparation of this paper All rights reserved by wwwijaresmnet ISSN :

14 REFERENCES [01] Agnes M, Buenaventura N, Labao JJ, Soria C K, Limbaco K A and Natividad LR, Inclusion of missing terms (Fibonacci mean) in a Fibonacci sequence, Math Investigatory Project, Central Luzon State University(2010) th [02] Bueno ACF, Solving the k term of Natividad s Fibonacci-like Sequence, International Journal of Mathematics and Scientific Computing (ISSN: ), Vol 3, No 1, 2013 [03] Shah D V, Special cases of Tribonacci periodicity, Journal of the Indian Academy of Mathematics, Vol 33, No2, 2011, [04] [4] Howell P, Nth term of the Fibonacci sequence, from Math Proofs: Interesting mathematical results and elegant solutions to various problems [05] Lin PY, De Moivre-type identities for the Tribonacci numbers, Fibonacci Quart 26 (2) (1988), [06] McCarty CP, A formula for Tribonacci numbers, Fibonacci Quart 19 (5) (1981), [07] Natividad L R, Deriving a Formula in Solving Fibonacci-like Sequence, International Journal of Mathematic and Scientific Computing, Vol 1, No 1, 2011 All rights reserved by wwwijaresmnet ISSN :

Extended Binet s formula for the class of generalized Fibonacci sequences

[VNSGU JOURNAL OF SCIENCE AND TECHNOLOGY] Vol4 No 1, July, 2015 205-210,ISSN : 0975-5446 Extended Binet s formula for the class of generalized Fibonacci sequences DIWAN Daksha M Department of Mathematics,