Sum of Product of Reciprocals of Fibonacci Numbers

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SUM OF PRODUCT OF RECIPROCALS OF FIBONACCI NUMBERS

A Thesis Submitted in the Partial Fulfillment of the Requirements of Degree for

Integrated M.Sc.

In Mathematics

Submitted by

Kappagantu Prudhavi Nag Roll Number: 410MA5016

Under the Guidance of Professor G. K. Panda Department of Mathematics

National Institute of Technology, Rourkela

May 2015

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CERTIFICATE

Dr. Gopal Krishna Panda

Professor of Mathematics May 11, 2015

This is to certify that the project report with title β€œSUM OF PRODUCT OF RECIPROCALS OF FIBONACCI NUMBERS” submitted by Mr. Kappagantu Prudhavi Nag, Roll No. 410MA5016, to the National Institute of Technology, Rourkela, Odisha for the partial fulfillment of the requirements of Integrated M.Sc. degree in Mathematics, is a bonafide research work carried out by him under my supervision and guidance. The content of this report in full or part has not been submitted to any other Institute or University for the award of any degree or diploma.

Gopal Krishna Panda

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ACKNOWLEDGEMENTS

I would like to express my special appreciation and thanks to my supervisor Professor G. K. Panda who has been a great mentor for me. I thank him for his suggestion for providing a beautiful problem in number theory.

I am grateful to Prof. Sunil Kumar Sarangi, Director, National Institute of Technology, Rourkela for providing excellent facilities in the Institute for carrying out research.

I am thankful to the Head, mathematics and professors of department for their valuable help during the preparation of this work.

There are no words to explain how grateful I am to my parents for all of their sacrifices and prayers that they made for me. I also like to thank my brother who helped me in this research and also gave a few ideas which helped me in successful completion of this project.

I also take this opportunity to thank all of my friends who helped me during the preparation of this work.

Place: Rourkela

Date: Kappagantu Prudhavi Nag

Department of Mathematics

National Institute of Technology Rourkela

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ABSTRACT

Fibonacci numbers are the number sequences which follow the linear mathematical recurrence𝐹0 = 0, 𝐹1 = 1 and 𝐹𝑛 = πΉπ‘›βˆ’1+ πΉπ‘›βˆ’2 𝑛 β‰₯ 2. In this work, we study certain sum formulas involving products of reciprocals of Fibonacci numbers. Sum formulas with alternating signs are also studied.

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TABLE OF CONTENTS

1. Notations ...6

2. Introduction ...7

i. Mathematics of Fibonacci Numbers ... 7

ii. Fibonacci Numbers with Negative Indices ... 8

3. Sum of Reciprocals of Fibonacci Numbers with Positive Indices ... 11

i. Order 2 ... 11

a. Non-Alternating Sum... 11

b. Alternating Sum ... 13

ii. Sum With Indices in A.P. ... 16

a. Non-Alternating Sum of Order 1 ... 16

b. Alternating Sum of Order 1 ... 16

4. Sum of Reciprocals of Fibonacci Numbers with Negative Indices ... 19

i. Order 1 ... 19

a. Non-Alternating Sum... 20

b. Alternating Sum ... 20

ii. Order 2 ... 20

a. Non-Alternating Sum... 21

b. Alternating Sum ... 21

iii. Sum with Indices in A.P. ... 22

a. Non-Alternating Sum of Order 1 ... 23

b. Alternating Sum of Order 1 ... 24

5. References ... 26

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NOTATIONS

The following notations will be frequently used in this thesis.

ο‚· 𝔽𝑁 = βˆ‘ 1

𝐹𝑛 𝑁𝑛=1

ο‚· 𝔾𝑁 = βˆ‘ (βˆ’1)𝑛

𝐹𝑛 𝑁𝑛=1

ο‚· 𝔽𝑁(π‘Ž) = βˆ‘ 1

𝐹𝑛+π‘Ž 𝑁𝑛=1

ο‚· 𝔾𝑁(π‘Ž) = βˆ‘ (βˆ’1)𝑛

𝐹𝑛+π‘Ž 𝑁𝑛=1

ο‚· π”½π‘β„Ž = βˆ‘ 1

πΉβ„Žπ‘› 𝑁𝑛=1

ο‚· π”Ύπ‘β„Ž = βˆ‘ (βˆ’1)𝑛

πΉβ„Žπ‘› 𝑁𝑛=1

ο‚· ℍ𝑁 = βˆ‘ 1

πΉβˆ’π‘› 𝑁𝑛=1

ο‚· 𝕀𝑁= βˆ‘ (βˆ’1)𝑛

πΉβˆ’π‘› 𝑁𝑛=1

ο‚· ℍ𝑁(π‘Ž) = βˆ‘ 1

πΉβˆ’π‘›βˆ’π‘Ž 𝑁𝑛=1

ο‚· 𝕀𝑁(π‘Ž) = βˆ‘ 1

πΉβˆ’π‘›βˆ’π‘Ž 𝑁𝑛=1

ο‚· ℍ𝑁(0, π‘Ž) = βˆ‘ 1

πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž 𝑁𝑛=1

ο‚· 𝕀𝑁(0, π‘Ž) = βˆ‘ (βˆ’1)𝑛

πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž 𝑁𝑛=1

ο‚· ℍ𝑁(π‘Ž, 𝑏) = βˆ‘ 1

πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘ 𝑁𝑛=1

ο‚· 𝕀𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛

πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘ 𝑁𝑛=1

ο‚· β„π‘β„Ž = βˆ‘ 1

πΉβˆ’β„Žπ‘› 𝑁𝑛=1

ο‚· π•€π‘β„Ž = βˆ‘ (βˆ’1)𝑛

πΉβˆ’β„Žπ‘› 𝑁𝑛=1

ο‚· β„π‘β„Ž(π‘Ž) = βˆ‘ 1

πΉβˆ’β„Žπ‘›βˆ’π‘Ž 𝑁𝑛=1

ο‚· π•€π‘β„Ž(π‘Ž) = βˆ‘ (βˆ’1)𝑛

πΉβˆ’β„Žπ‘›βˆ’π‘Ž 𝑁𝑛=1

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CHAPTER 1 INTRODUCTION

Fibonacci sequence of numbers is one of the most intriguing number sequence in mathematics. The series is named after the famous Italian Mathematician Fibonacci of the Bonacci family. He is also known as the Leonardo of Pisa. The following problem proposed by Fibonacci himself gave birth to the sequence.

The Fibonacci Problem:

Suppose there are two newborn rabbits, one male and one female.

Find the number of rabbits produced in a year if [1]

ο‚· Each pair takes one month to become mature

ο‚· Each pair produces a mixed pair every month from the second month

ο‚· All rabbits are immortal

Solution:

For convenience let us assume that the rabbits are born on January 1st and we need to find the number of rabbits on December 1st. The table below is used to find the solution of the problem.

From the above table it is evident that the number of rabbits at the end of the year are 144. If observed closely it is observed that the new number is equal to the sum of the previous two numbers.

MATHEMATICS OF FIBONACCI NUMBERS

The numbers in the bottom row are called the Fibonacci numbers. From the table a recursive relation is yielded as below

𝐹𝑛 = πΉπ‘›βˆ’1+ πΉπ‘›βˆ’2 , 𝑛 β‰₯ 2.

where 𝐹0 = 0 and 𝐹1 = 1 [2]. Sometimes it is customary to start the Fibonacci numbers from 𝐹1 instead of 𝐹0. Then the initial two conditions become 𝐹1 = 1 and 𝐹2 = 1. With any of the above two conditions the series generated is the same.

No. of Pairs Jan Feb Mar April May June July Aug Sep Oct Nov Dec

Adults 0 1 1 2 3 5 8 13 21 34 55 89

Babies 1 0 1 1 2 3 5 8 13 21 34 34

Total 1 1 2 3 5 8 13 21 34 55 89 144

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Page | 8 It is surprising that Fibonacci numbers can be extracted from Pascal’s triangle. The above observation was confirmed by Lucas in 1876 when he derived a straightforward formula to find the Fibonacci numbers [1]

𝐹𝑛 = βˆ‘ (𝑛 βˆ’ 𝑖 βˆ’ 1

𝑖 )

⌊(π‘›βˆ’1) 2⁄ βŒ‹

𝑖=0

, 𝑛 β‰₯ 1.

In 1843 a French mathematician named Jacques Philippe Marie Binet invented a way to calculate the π‘›π‘‘β„Ž Fibonacci numbers. If πœ™ =1+√5

2 and πœ“ =1βˆ’βˆš5

2 then 𝐹𝑛 = πœ™π‘›βˆ’πœ“π‘›

√5 [3].

The above formula shows an interesting aspect that the Fibonacci number can be written in terms of the golden ratio. Fibonacci numbers appear in many places in both nature and mathematics. They occur in music, geography, nature, and geometry. They can be found in the spiral arrangements of seeds of sunflowers, the scale patterns of pine cones, the arrangement of leaves and the number of petals on the flower.

FIBONACCI NUMBERS WITH NEGATIVE INDICES

The negative subscript of the Fibonacci numbers can be converted to positive subscript as indicated in [4]. The recurrence relation for the negative Fibonacci numbers is as follows:

πΉβˆ’π‘› = 𝐹2βˆ’π‘›βˆ’ 𝐹1βˆ’π‘›, 𝑛 β‰₯ 2

The initial conditions for these numbers are 𝐹0 = 0 and 𝐹1 = 1. It is observed that the negative Fibonacci numbers have the same initial conditions as of the positive Fibonacci numbers.

Rabinowitz [5] stated that the alternating general summation of order 1 is given by

𝔾𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛 𝐹𝑛+π‘Ž

𝑁

𝑛=1

= 𝔾𝑁+π‘Ž βˆ’ π”Ύπ‘Ž.

We disprove the above identity using a counter example.

Now let us take π‘Ž = 3 and 𝑁 = 5 and find the value of 𝔾𝑁(π‘Ž).

βˆ‘(βˆ’1)𝑛

𝐹𝑛+3

5

𝑛=1

=βˆ’1

𝐹4 + 1 𝐹5+βˆ’1

𝐹6 + 1 𝐹7+βˆ’1

𝐹8 𝔾8 =βˆ’1

𝐹1 + 1 𝐹2+βˆ’1

𝐹3 + 1 𝐹4+βˆ’1

𝐹5 + 1 𝐹6+βˆ’1

𝐹7 + 1 𝐹8

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Page | 9 𝔾3 =βˆ’1

𝐹1 + 1 𝐹2+βˆ’1

𝐹3 𝔾𝑁+π‘Žβˆ’ π”Ύπ‘Ž= 1

𝐹4+βˆ’1 𝐹5 + 1

𝐹6+βˆ’1 𝐹7 + 1

𝐹8β‰  𝔾𝑁(π‘Ž).

It can be checked that the identity for 𝔾𝑁(π‘Ž) is wrong when π‘Ž is odd and correct when π‘Ž is even.

Claim:

If π‘Ž > 0 then 𝔾𝑁(π‘Ž) = {π”Ύπ‘Žβˆ’ 𝔾𝑁+π‘Ž if π‘Ž is odd, 𝔾𝑁+π‘Žβˆ’ π”Ύπ‘Ž if π‘Ž is even.

Proof:

𝔾𝑁+π‘Žβˆ’ π”Ύπ‘Ž = βˆ‘(βˆ’1)𝑛 𝐹𝑛

𝑁+π‘Ž

𝑛=1

βˆ’ βˆ‘(βˆ’1)𝑛 𝐹𝑛

π‘Ž

𝑛=1

= βˆ‘ (βˆ’1)𝑛 𝐹𝑛

𝑁+π‘Ž

𝑛=π‘Ž+1

.

Case 1. π‘Ž is odd, 𝑁 is odd

𝔾𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛 𝐹𝑛+π‘Ž

𝑁

𝑛=1

= βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)+π‘Ž+ βˆ’1 𝐹𝑁+π‘Ž. 𝔾𝑁+π‘Ž βˆ’ π”Ύπ‘Ž = βˆ‘ (βˆ’1)𝑛

𝐹𝑛

𝑁+π‘Ž

𝑛=π‘Ž+1

= 1

𝐹1+𝑏 + βˆ’1

𝐹2+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)+π‘Ž+ 1 𝐹𝑁+π‘Ž

= βˆ’ ( βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)+π‘Ž + βˆ’1 𝐹𝑁+π‘Ž)

= βˆ’π”Ύπ‘(π‘Ž).

Case 2. π‘Ž is odd, 𝑁 is even

𝔾𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛 𝐹𝑛+π‘Ž

𝑁

𝑛=1

= βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)+π‘Ž+ 1 𝐹𝑁+π‘Ž.

𝔾𝑁+π‘Ž βˆ’ 𝔾𝑏= βˆ‘ (βˆ’1)𝑛 𝐹𝑛

𝑁+π‘Ž

𝑛=π‘Ž+1

= 1

𝐹1+π‘Ž+ βˆ’1

𝐹2+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)+π‘Ž+ βˆ’1 𝐹𝑁+π‘Ž

= βˆ’ ( βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)+π‘Ž+ 1 𝐹𝑁+π‘Ž)

= βˆ’π”Ύπ‘(π‘Ž).

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Page | 10 Case 3. π‘Ž is even , 𝑁 is odd

𝔾𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛 𝐹𝑛+π‘Ž

𝑁

𝑛=1

= βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)+π‘Ž+ βˆ’1 𝐹𝑁+π‘Ž. 𝔾𝑁+π‘Žβˆ’ 𝔾𝑏 = βˆ‘ (βˆ’1)𝑛

𝐹𝑛

𝑁+π‘Ž

𝑛=π‘Ž+1

= βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)+π‘Ž+ βˆ’1 𝐹𝑁+π‘Ž

= 𝔾𝑁(π‘Ž).

Case 4. π‘Ž is even , 𝑁 is even

𝔾𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛 𝐹𝑛+π‘Ž

𝑁

𝑛=1

= βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)+π‘Ž+ 1 𝐹𝑁+π‘Ž. 𝔾𝑁+π‘Žβˆ’ 𝔾𝑏 = βˆ‘ (βˆ’1)𝑛

𝐹𝑛

𝑁+π‘Ž

𝑛=π‘Ž+1

= βˆ’1 𝐹1+π‘Ž+ 1

𝐹2+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)+π‘Ž+ 1 𝐹𝑁+π‘Ž

= 𝔾𝑁(π‘Ž).

∎

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CHAPTER 2

SUM OF RECIPROCALS OF FIBONACCI NUMBERS WITH POSITIVE INDICES

The following notations for the alternating and non-alternating sum of π‘˜π‘‘β„Ž order are available in[5].

𝑆(π‘Ž1, π‘Ž2, … , π‘Žπ‘˜βˆ’1, π‘Žπ‘˜ ) = βˆ‘ 1 𝐹𝑛+π‘Ž1𝐹𝑛+π‘Ž2…𝐹

𝑛+π‘Žπ‘˜ 𝑁

𝑛=1

𝑇(π‘Ž1, π‘Ž2, … , π‘Žπ‘˜βˆ’1, π‘Žπ‘˜ ) = βˆ‘ (βˆ’1)𝑛 𝐹𝑛+π‘Ž1𝐹𝑛+π‘Ž2…𝐹

𝑛+π‘Žπ‘˜ 𝑁

𝑛=1

The sum 𝑆 is called the non-alternating summation of order π‘˜. The second sum 𝑇 is called the alternating sum of order π‘˜. In both the cases 0 < π‘Ž1 < π‘Ž2 < β‹― < π‘Žπ‘˜βˆ’1 < π‘Žπ‘˜.

ORDER 2

We consider first the problem of finding the following second order sums.

𝔽𝑁(π‘Ž, 𝑏 ) = βˆ‘ 1 𝐹𝑛+π‘ŽπΉπ‘›+𝑏

𝑁

𝑛=1

𝔾𝑁(π‘Ž, 𝑏 ) = βˆ‘ (βˆ’1)𝑛 𝐹𝑛+π‘ŽπΉπ‘›+𝑏

𝑁

𝑛=1

NON-ALTERNATING SUM

For π‘Ž > 0, Rabinowitz[5] got the following formula.

𝔽𝑁(0, π‘Ž) = βˆ‘ 1 𝐹𝑛𝐹𝑛+π‘Ž =

{ 1

πΉπ‘Ž βˆ‘ ( 1

𝐹𝑁+2𝑖𝐹𝑁+2𝑖+1βˆ’ 1 𝐹2𝑖𝐹2𝑖+1)

βŒŠπ‘Ž 2⁄ βŒ‹

𝑖=1

+𝕂𝑁

πΉπ‘Ž if π‘Ž is odd, 1

πΉπ‘Žβˆ‘ ( 1

𝐹2π‘–βˆ’1𝐹2𝑖 βˆ’ 1

𝐹𝑁+2π‘–βˆ’1𝐹𝑁+2𝑖)

π‘Ž 2⁄

𝑖=1

if π‘Ž is even.

𝑁

𝑛=1

where 𝕂𝑁= βˆ‘ 1

𝐹𝑛𝐹𝑛+1

𝑁𝑖=1 . The above sum is called the non-alternating sum of order 2. The aim is to find an equivalent expression of 𝔽𝑁(π‘Ž, 𝑏). To do this we use the help of the following result.

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Page | 12 π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š 1: πΉπ‘œπ‘Ÿ 0 < π‘Ž < 𝑏 π‘‘β„Žπ‘’π‘› 𝐹𝑁 (π‘Ž, 𝑏) = 𝐹𝑁+π‘Ž (0, 𝑏 βˆ’ π‘Ž) βˆ’ πΉπ‘Ž (0, 𝑏 βˆ’ π‘Ž).

π‘ƒπ‘Ÿπ‘œπ‘œπ‘“:We start from the right hand side of the equation and come to the left hand side.

𝔽𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”½π‘Ž(0, 𝑏 βˆ’ π‘Ž) = βˆ‘ 1

𝐹𝑛𝐹𝑛+π‘βˆ’π‘Žβˆ’ βˆ‘ 1 𝐹𝑛𝐹𝑛+π‘βˆ’π‘Ž

π‘Ž

𝑖=1 𝑁+π‘Ž

𝑛=1

= βˆ‘ 1

𝐹𝑛𝐹𝑛+π‘βˆ’π‘Ž

𝑁+π‘Ž

𝑛=π‘Ž+1

= 1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + 1

𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1+ 1 𝐹𝑁+π‘ŽπΉπ‘+𝑏

= βˆ‘ 1

𝐹𝑛+π‘ŽπΉπ‘›+𝑏

𝑁

𝑛=1

= 𝔽𝑁(π‘Ž, 𝑏)

∎ By using the above theorem and the formula for 𝔽𝑁(0, π‘Ž) stated by Rabinowitz [5] the expression for 𝔽𝑁(π‘Ž, 𝑏) is calculated.

π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š 2: 𝐼𝑓 0 < π‘Ž < 𝑏 π‘‘β„Žπ‘’π‘›

𝔽𝑁(π‘Ž, 𝑏) = { 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹𝑁+π‘Ž+2𝑖𝐹𝑁+π‘Ž+2𝑖+1βˆ’ 1 πΉπ‘Ž+2π‘–πΉπ‘Ž+2𝑖+1)

⌊(π‘βˆ’π‘Ž) 2⁄ βŒ‹

𝑖=1

+𝔽𝑁(π‘Ž, π‘Ž + 1)

πΉπ‘βˆ’π‘Ž if (𝑏 βˆ’ π‘Ž)is odd,

1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

πΉπ‘Ž+2π‘–βˆ’1πΉπ‘Ž+2π‘–βˆ’ 1

𝐹𝑁+π‘Ž+2π‘–βˆ’1𝐹𝑁+π‘Ž+2𝑖)

(π‘βˆ’π‘Ž) 2⁄

𝑖=1

if (𝑏 βˆ’ π‘Ž)is even.

π‘ƒπ‘Ÿπ‘œπ‘œπ‘“

:

We take the help of Theorem 1 to prove this theorem.

𝔽𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) =

{ 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹𝑁+π‘Ž+2𝑖𝐹𝑁+π‘Ž+2𝑖+1βˆ’ 1 𝐹2𝑖𝐹2𝑖+1)

⌊(π‘βˆ’π‘Ž) 2⁄ βŒ‹

𝑖=1

+𝕂𝑁+π‘Ž

πΉπ‘βˆ’π‘Ž if (𝑏 βˆ’ π‘Ž) is odd, 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹2π‘–βˆ’1𝐹2π‘–βˆ’ 1

𝐹𝑁+π‘Ž+2π‘–βˆ’1𝐹𝑁+π‘Ž+2𝑖)

𝑏(π‘βˆ’π‘Ž) 2⁄ βˆ’π‘Ž

⁄2

𝑖=1

if (𝑏 βˆ’ π‘Ž) is even.

π”½π‘Ž(0, 𝑏 βˆ’ π‘Ž) = { 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

πΉπ‘Ž+2π‘–πΉπ‘Ž+2𝑖+1βˆ’ 1 𝐹2𝑖𝐹2𝑖+1)

⌊(π‘βˆ’π‘Ž) 2⁄ βŒ‹

𝑖=1

+ π•‚π‘Ž

πΉπ‘βˆ’π‘Ž if (𝑏 βˆ’ π‘Ž) is odd, 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹2π‘–βˆ’1𝐹2π‘–βˆ’ 1

πΉπ‘Ž+2π‘–βˆ’1πΉπ‘Ž+2𝑖)

(π‘βˆ’π‘Ž) 2⁄

𝑖=1

if (𝑏 βˆ’ π‘Ž) is even.

Depending upon the parity of (𝑏 βˆ’ π‘Ž), two different cases are taken into consideration

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Page | 13 Case 1. (𝒃 βˆ’ 𝒂) 𝐒𝐬 𝐨𝐝𝐝

In this case,

𝔽𝑁(π‘Ž, 𝑏) = 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹𝑁+π‘Ž+2𝑖𝐹𝑁+π‘Ž+2𝑖+1βˆ’ 1

𝐹2𝑖𝐹2𝑖+1βˆ’ 1

πΉπ‘Ž+2π‘–πΉπ‘Ž+2𝑖+1+ 1 𝐹2𝑖𝐹2𝑖+1)

⌊(π‘βˆ’π‘Ž) 2⁄ βŒ‹

𝑖=1

+𝕂𝑁+π‘Žβˆ’ π•‚π‘Ž πΉπ‘βˆ’π‘Ž

= 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹𝑁+π‘Ž+2𝑖𝐹𝑁+π‘Ž+2𝑖+1βˆ’ 1

πΉπ‘Ž+2π‘–πΉπ‘Ž+2𝑖+1) +𝕂𝑁+π‘Žβˆ’ π•‚π‘Ž πΉπ‘βˆ’π‘Ž .

⌊(π‘βˆ’π‘Ž) 2⁄ βŒ‹

𝑖=1

Now,

𝕂𝑁+π‘Ž βˆ’ 𝕂𝑁 = βˆ‘ 1 𝐹𝑛𝐹𝑛+1

𝑁+π‘Ž

𝑛=1

βˆ’ βˆ‘ 1

𝐹𝑛𝐹𝑛+1

π‘Ž

𝑛=1

= βˆ‘ 1

𝐹𝑛𝐹𝑛+1

𝑁+π‘Ž

𝑛=π‘Ž+1

= βˆ‘ 1

𝐹𝑛+π‘ŽπΉπ‘›+π‘Ž+1= 𝔽𝑁(π‘Ž, π‘Ž + 1)

𝑁

𝑛=1

𝔽𝑁(π‘Ž, 𝑏) = 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹𝑁+π‘Ž+2𝑖𝐹𝑁+π‘Ž+2𝑖+1βˆ’ 1 πΉπ‘Ž+2π‘–πΉπ‘Ž+2𝑖+1)

⌊(π‘βˆ’π‘Ž) 2⁄ βŒ‹

𝑖=1

+𝔽𝑁(π‘Ž, π‘Ž + 1) πΉπ‘βˆ’π‘Ž . Case 2. (𝑏 βˆ’ π‘Ž) is even

𝔽𝑁(π‘Ž, 𝑏) = 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

𝐹2π‘–βˆ’1𝐹2π‘–βˆ’ 1

𝐹𝑁+π‘Ž+2π‘–βˆ’1𝐹𝑁+π‘Ž+2π‘–βˆ’ 1

𝐹2π‘–βˆ’1𝐹2π‘–βˆ’ 1

πΉπ‘Ž+2π‘–βˆ’1πΉπ‘Ž+2𝑖)

(π‘βˆ’π‘Ž) 2⁄

𝑖=1

= 1

πΉπ‘βˆ’π‘Ž βˆ‘ ( 1

πΉπ‘Ž+2π‘–βˆ’1πΉπ‘Ž+2π‘–βˆ’ 1

𝐹𝑁+π‘Ž+2π‘–βˆ’1𝐹𝑁+π‘Ž+2𝑖)

(π‘βˆ’π‘Ž) 2⁄

𝑖=1

.

∎

ALTERNATING SUM

Let π‘Ž > 0 . The following identity is available in from [6].

𝔾𝑁(0, π‘Ž) = βˆ‘ (βˆ’1)𝑛 𝐹𝑛𝐹𝑛+π‘Ž = 1

πΉπ‘Žβˆ‘ (πΉπ‘—βˆ’1

𝐹𝑗 βˆ’πΉπ‘+π‘—βˆ’1 𝐹𝑁+𝑗 )

π‘Ž

𝑗=1 𝑁

𝑛=1

.

The above is a sum with alternating sign (called an alternating sum) of order 2. We use the above sum to find a formula for 𝔾𝑁(π‘Ž, 𝑏). To achieve this, we use of the following result.

π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š 3: 𝐼𝑓 0 < π‘Ž < 𝑏 π‘‘β„Žπ‘’π‘›

𝔾𝑁(π‘Ž, 𝑏) = {π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ 𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) if π‘Ž is odd, 𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) if π‘Ž is even.

π‘ƒπ‘Ÿπ‘œπ‘œπ‘“: Observe that

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Page | 14 𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) = βˆ‘ (βˆ’1)𝑛

𝐹𝑁𝐹𝑛+π‘βˆ’π‘Ž

𝑁+π‘Ž

𝑛=1

βˆ’ βˆ‘ (βˆ’1)𝑛 𝐹𝑁𝐹𝑛+π‘βˆ’π‘Ž

π‘Ž

𝑛=1

= βˆ‘ (βˆ’1)𝑛 𝐹𝑛𝐹𝑛+π‘βˆ’π‘Ž.

𝑁+π‘Ž

𝑛=π‘Ž+1

We distinguish four cases:

Case 1. π‘Ž and 𝑁 are odd 𝔾𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛

𝐹𝑛+π‘ŽπΉπ‘›+𝑏 = βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + 1 𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1

𝑁

𝑛=1

+ βˆ’1

𝐹𝑁+π‘ŽπΉπ‘+𝑏.

𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) = βˆ‘ (βˆ’1)𝑛 𝐹𝑛𝐹𝑛+π‘βˆ’π‘Ž

𝑁+π‘Ž

𝑛=π‘Ž+1

= 1

πΉπ‘Ž+1𝐹𝑏+2+ βˆ’1

πΉπ‘Ž+2𝐹𝑏+3+ β‹― + βˆ’1

𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1+ 1 𝐹𝑁+π‘ŽπΉπ‘+𝑏

= βˆ’ ( βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + 1

𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1+ βˆ’1 𝐹𝑁+π‘ŽπΉπ‘+𝑏)

= βˆ’π”Ύπ‘(π‘Ž, 𝑏).

Case 2. π‘Ž is odd and 𝑁 is even 𝔾𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛

𝐹𝑛+π‘ŽπΉπ‘›+𝑏 = βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + βˆ’1 𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1

𝑁

𝑛=1

+ 1

𝐹𝑁+π‘ŽπΉπ‘+𝑏.

𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) = βˆ‘ (βˆ’1)𝑛 𝐹𝑛𝐹𝑛+π‘βˆ’π‘Ž

𝑁+π‘Ž

𝑛=π‘Ž+1

= 1

πΉπ‘Ž+1𝐹𝑏+2+ βˆ’1

πΉπ‘Ž+2𝐹𝑏+3+ β‹― + 1

𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1+ βˆ’1 𝐹𝑁+π‘ŽπΉπ‘+𝑏

= βˆ’ ( βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + βˆ’1

𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1++ 1 𝐹𝑁+π‘ŽπΉπ‘+𝑏)

= βˆ’π”Ύπ‘(π‘Ž, 𝑏).

Case 3. π‘Ž is even and 𝑁 is odd 𝔾𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛

𝐹𝑛+π‘ŽπΉπ‘›+𝑏 = βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + 1 𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1

𝑁

𝑛=1

+ βˆ’1

𝐹𝑁+π‘ŽπΉπ‘+𝑏.

𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) = βˆ‘ (βˆ’1)𝑛 𝐹𝑛𝐹𝑛+π‘βˆ’π‘Ž

𝑁+π‘Ž

𝑛=π‘Ž+1

= βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + 1

𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1++ βˆ’1 𝐹𝑁+π‘ŽπΉπ‘+𝑏

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Page | 15

= 𝔾𝑁(π‘Ž, 𝑏).

Case 4. π‘Ž and 𝑁 are even 𝔾𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛

𝐹𝑛+π‘ŽπΉπ‘›+𝑏 = βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + βˆ’1 𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1

𝑁

𝑛=1

+ 1

𝐹𝑁+π‘ŽπΉπ‘+𝑏

𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) = βˆ‘ (βˆ’1)𝑛 𝐹𝑛𝐹𝑛+π‘βˆ’π‘Ž

𝑁+π‘Ž

𝑛=π‘Ž+1

= βˆ’1

πΉπ‘Ž+1𝐹𝑏+1+ 1

πΉπ‘Ž+2𝐹𝑏+2+ β‹― + βˆ’1

𝐹𝑁+π‘Žβˆ’1𝐹𝑁+π‘βˆ’1++ 1 𝐹𝑁+π‘ŽπΉπ‘+𝑏

= 𝔾𝑁(π‘Ž, 𝑏).

∎ The following theorem is one of the main results of this chapter.

π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š 4: 𝐼𝑓 0 < π‘Ž < 𝑏 π‘‘β„Žπ‘’π‘›

𝔾𝑁(π‘Ž, 𝑏) = {

1

πΉπ‘βˆ’π‘Žβˆ‘ (𝐹𝑁+π‘Ž+π‘—βˆ’1

𝐹𝑁+π‘Ž+𝑗 βˆ’πΉπ‘Ž+π‘—βˆ’1 πΉπ‘Ž+𝑗 )

π‘βˆ’π‘Ž

𝑗=1

if π‘Ž is odd, 1

πΉπ‘βˆ’π‘Žβˆ‘ (πΉπ‘Ž+π‘—βˆ’1

πΉπ‘Ž+𝑗 βˆ’πΉπ‘+π‘Ž+π‘—βˆ’1 𝐹𝑁+π‘Ž+𝑗 )

π‘βˆ’π‘Ž

𝑗=1

if π‘Ž is even.

π‘ƒπ‘Ÿπ‘œπ‘œπ‘“: We separate two cases:

Case 1. π‘Ž is odd

π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ 𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) = 1

πΉπ‘βˆ’π‘Žβˆ‘ ( πΉπ‘—βˆ’1

𝐹𝑗 βˆ’πΉπ‘Ž+π‘—βˆ’1 πΉπ‘Ž+𝑗 )

π‘βˆ’π‘Ž

𝑗=1

βˆ’ 1

πΉπ‘βˆ’π‘Žβˆ‘ ( πΉπ‘—βˆ’1

𝐹𝑗 βˆ’πΉπ‘+π‘Ž+π‘—βˆ’1 𝐹𝑁+π‘Ž+𝑗 )

π‘βˆ’π‘Ž

𝑗=1

= 1

πΉπ‘βˆ’π‘Žβˆ‘ (

𝐹𝑁+π‘Ž+π‘—βˆ’1

𝐹𝑁+π‘Ž+𝑗 βˆ’πΉπ‘Ž+π‘—βˆ’1 πΉπ‘Ž+𝑗 )

π‘βˆ’π‘Ž

𝑗=1

. Case 2. π‘Ž is even

𝔾𝑁+π‘Ž(0, 𝑏 βˆ’ π‘Ž) βˆ’ π”Ύπ‘Ž(0, 𝑏 βˆ’ π‘Ž) = 1

πΉπ‘βˆ’π‘Žβˆ‘ ( πΉπ‘—βˆ’1

𝐹𝑗 βˆ’πΉπ‘+π‘Ž+π‘—βˆ’1 𝐹𝑁+π‘Ž+𝑗 )

π‘βˆ’π‘Ž

𝑗=1

βˆ’ 1

πΉπ‘βˆ’π‘Žβˆ‘ ( πΉπ‘—βˆ’1

𝐹𝑗 βˆ’πΉπ‘Ž+π‘—βˆ’1 πΉπ‘Ž+𝑗 )

π‘βˆ’π‘Ž

𝑗=1

= 1

πΉπ‘βˆ’π‘Žβˆ‘ ( πΉπ‘Ž+π‘—βˆ’1

πΉπ‘Ž+𝑗 βˆ’πΉπ‘+π‘Ž+π‘—βˆ’1 𝐹𝑁+π‘Ž+𝑗 ).

π‘βˆ’π‘Ž

𝑗=1

∎

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Page | 16

SUM WITH INDICES IN A.P.

Let β„Ž be a natural number. We express certain sums in terms of the following two sums.

π”½π‘β„Ž = βˆ‘ 1 πΉβ„Žπ‘›

𝑁

𝑛=1

, π”Ύπ‘β„Ž = βˆ‘(βˆ’1)𝑛 πΉβ„Žπ‘›

𝑁

𝑛=1

NON-ALTERNATING SUM OF ORDER 1

We consider the problem of finding π”½π‘β„Ž(π‘Ž) in terms of π”½π‘β„Ž. π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š 5: 𝐼𝑓 π‘Ž > 0 π‘Žπ‘›π‘‘ β„Ž|π‘Ž π‘‘β„Žπ‘’π‘› π”½π‘β„Ž(π‘Ž) = 𝔽𝑁+π‘Ž

β„Ž β„Ž βˆ’ π”½π‘Ž

β„Ž β„Ž .

π‘ƒπ‘Ÿπ‘œπ‘œπ‘“:Let 𝑏 =π‘Ž

β„Ž and π‘Ž = π‘β„Ž then, 𝔽𝑁+π‘Ž

β„Ž β„Ž βˆ’ π”½π‘Ž

β„Ž

β„Ž= βˆ‘ 1

πΉβ„Žπ‘›

𝑁+𝑏

𝑛=1

βˆ’ βˆ‘ 1

πΉβ„Žπ‘›

𝑏

𝑛=1

= βˆ‘ 1

πΉβ„Žπ‘›

𝑁+𝑏

𝑛=𝑏+1

= 1

πΉβ„Ž(𝑏+1)+ 1

πΉβ„Ž(𝑏+2)+ β‹― + 1

πΉβ„Ž(𝑁+π‘βˆ’1)+ 1 πΉβ„Ž(𝑁+𝑏)

= 1

πΉβ„Ž+π‘Ž+ 1

𝐹2β„Ž+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ 1 πΉπ‘β„Ž+π‘Ž

= π”½π‘β„Ž(π‘Ž).

∎

ALTERNATING SUM OF ORDER 1

We consider the problem of finding π”Ύπ‘β„Ž(π‘Ž) in terms of π”Ύπ‘β„Ž. π‘‡β„Žπ‘’π‘œπ‘Ÿπ‘’π‘š 6: 𝐼𝑓 π‘Ž > 0 π‘Žπ‘›π‘‘ β„Ž|π‘Ž π‘‘β„Žπ‘’π‘›

π”Ύπ‘β„Ž(π‘Ž) = { π”Ύπ‘Ž

β„Ž β„Žβˆ’ 𝔾

𝑁+π‘Ž β„Ž

β„Ž if π‘Ž is odd, 𝔾𝑁+π‘Ž

β„Ž β„Ž βˆ’ π”Ύπ‘Ž

β„Ž

β„Ž if π‘Ž is even.

Proof:

Let 𝑏 =π‘Ž

β„Ž and π‘Ž = π‘β„Ž then,

𝔾𝑁+π‘β„Ž βˆ’ π”Ύπ‘β„Ž = βˆ‘(βˆ’1)𝑛 πΉβ„Žπ‘›

𝑁+𝑏

𝑛=1

βˆ’ βˆ‘(βˆ’1)𝑛 πΉβ„Žπ‘›

𝑏

𝑛=1

= βˆ‘ (βˆ’1)𝑛 πΉβ„Žπ‘›

𝑁+𝑏

𝑛=𝑏+1

.

Once again we distinguish four cases.

Case 1. 𝑏 =π‘Ž

β„Ž and 𝑁 are odd.

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Page | 17 π”Ύπ‘β„Ž(π‘Ž) = βˆ‘(βˆ’1)𝑛

πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= βˆ’1

πΉβ„Ž+π‘Ž+ 1

𝐹2β„Ž+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ βˆ’1 πΉπ‘β„Ž+π‘Ž. 𝔾𝑁+π‘β„Ž βˆ’ π”Ύπ‘β„Ž = βˆ‘ (βˆ’1)𝑛

πΉβ„Žπ‘›

𝑁+𝑏

𝑛=𝑏+1

= 1

πΉβ„Ž+β„Žπ‘+ βˆ’1

𝐹2β„Ž+β„Žπ‘+ β‹― + βˆ’1

𝐹(π‘βˆ’1)β„Ž+β„Žπ‘ + 1 πΉπ‘β„Ž+β„Žπ‘

= 1

πΉβ„Ž+π‘Ž+ βˆ’1

𝐹2β„Ž+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ βˆ’1 πΉπ‘β„Ž+π‘Ž

= βˆ’ ( βˆ’1

πΉβ„Ž+π‘Ž+ 1

𝐹2β„Ž+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ βˆ’1 πΉπ‘β„Ž+π‘Ž)

= βˆ’π”Ύπ‘β„Ž(π‘Ž).

Case 2. 𝑏 =π‘Ž

β„Ž is odd, 𝑁 is even π”Ύπ‘β„Ž(π‘Ž) = βˆ‘(βˆ’1)𝑛

πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= βˆ’1

πΉβ„Ž+π‘Ž+ 1

𝐹2β„Ž+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ 1 πΉπ‘β„Ž+π‘Ž.

𝔾𝑁+π‘β„Ž βˆ’ π”Ύπ‘β„Ž = βˆ‘ (βˆ’1)𝑛 πΉβ„Žπ‘›

𝑁+𝑏

𝑛=𝑏+1

= 1

πΉβ„Ž+β„Žπ‘+ βˆ’1

𝐹2β„Ž+β„Žπ‘+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+β„Žπ‘ + βˆ’1 πΉπ‘β„Ž+β„Žπ‘

= 1

πΉβ„Ž+π‘Ž+ βˆ’1

𝐹2β„Ž+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ βˆ’1 πΉπ‘β„Ž+π‘Ž

= βˆ’ ( βˆ’1

πΉβ„Ž+π‘Ž+ 1

𝐹2β„Ž+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ βˆ’1 πΉπ‘β„Ž+π‘Ž)

= βˆ’π”Ύπ‘β„Ž(π‘Ž) Case 3. 𝑏 =π‘Ž

β„Ž is even , 𝑁 is odd π”Ύπ‘β„Ž(π‘Ž) = βˆ‘(βˆ’1)𝑛 πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= βˆ’1

πΉβ„Ž+π‘Ž+ 1

𝐹2β„Ž+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ βˆ’1 πΉπ‘β„Ž+π‘Ž. 𝔾𝑁+π‘β„Ž βˆ’ π”Ύπ‘β„Ž = βˆ‘ (βˆ’1)𝑛

πΉβ„Žπ‘›

𝑁+𝑏

𝑛=𝑏+1

= βˆ’1

πΉβ„Ž+β„Žπ‘+ 1

𝐹2β„Ž+β„Žπ‘+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+β„Žπ‘+ βˆ’1 πΉπ‘β„Ž+β„Žπ‘

= 1

πΉβ„Ž+π‘Ž+ βˆ’1

𝐹2β„Ž+π‘Ž+ β‹― + 1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ βˆ’1 πΉπ‘β„Ž+π‘Ž

= π”Ύπ‘β„Ž(π‘Ž).

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Page | 18 Case 4. 𝑏 = π‘Ž

β„Ž and 𝑁 are even π”Ύπ‘β„Ž(π‘Ž) = βˆ‘(βˆ’1)𝑛

πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= βˆ’1

πΉβ„Ž+π‘Ž+ 1

𝐹2β„Ž+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ 1 πΉπ‘β„Ž+π‘Ž. 𝔾𝑁+π‘β„Ž βˆ’ π”Ύπ‘β„Ž = βˆ‘ (βˆ’1)𝑛

πΉβ„Žπ‘›

𝑁+𝑏

𝑛=𝑏+1

= βˆ’1

πΉβ„Ž+β„Žπ‘+ 1

𝐹2β„Ž+β„Žπ‘+ β‹― + βˆ’1

𝐹(π‘βˆ’1)β„Ž+β„Žπ‘+ 1 πΉπ‘β„Ž+β„Žπ‘

= 1

πΉβ„Ž+π‘Ž+ βˆ’1

𝐹2β„Ž+π‘Ž+ β‹― + βˆ’1

𝐹(π‘βˆ’1)β„Ž+π‘Ž+ 1 πΉπ‘β„Ž+π‘Ž

= π”Ύπ‘β„Ž(π‘Ž).

∎

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Page | 19

CHAPTER 3

SUM OF RECIPROCALS OF FIBONACCI NUMBERS WITH NEGATIVE INDICES

We use the conversion of negative Fibonacci numbers with negative indices to Fibonacci numbers with positive indices and then derive the identities for these numbers. We use the following notations.

ℍ𝑁 = βˆ‘ 1 πΉβˆ’π‘›

𝑁

𝑛=1

, 𝕀𝑁= βˆ‘(βˆ’1)𝑛 πΉβˆ’π‘›

𝑁

𝑛=1

We first write them in terms of 𝔽𝑁 and 𝔾𝑁. We use the formula πΉβˆ’π‘›= (βˆ’1)𝑛+1𝐹𝑛

stated in [3]. Then we have

ℍ𝑁 = βˆ‘ 1 πΉβˆ’π‘›

𝑁

𝑛=1

= βˆ‘ 1

(βˆ’1)𝑛+1𝐹𝑛 =

𝑁

𝑛=1

βˆ‘(βˆ’1)𝑛+1 𝐹𝑛

𝑁

𝑛=1

= βˆ’ βˆ‘(βˆ’1)𝑛 𝐹𝑛

𝑁

𝑛=1

= βˆ’π”Ύπ‘,

𝕀𝑁 = βˆ‘(βˆ’1)𝑛 πΉβˆ’π‘›

𝑁

𝑛=1

= βˆ‘ (βˆ’1)𝑛 (βˆ’1)𝑛+1𝐹𝑛 =

𝑁

𝑛=1

βˆ‘(βˆ’1)1 𝐹𝑛

𝑁

𝑛=1

= βˆ’ βˆ‘ 1 𝐹𝑛

𝑁

𝑛=1

= βˆ’π”½π‘.

∎

ORDER 1

We first consider the following sums:

ℍ𝑁(π‘Ž) = βˆ‘ 1 πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

,

𝕀𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛 πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

.

We write the above in terms of 𝔽𝑁(π‘Ž) and 𝔾𝑁(π‘Ž).

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Page | 20

NON-ALTERNATING SUM

The non-alternating sum of Fibonacci numbers of negative indices of 1st order is,

ℍ𝑁(π‘Ž) = βˆ‘ 1 πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

. We write this sum in terms of 𝔾𝑁(π‘Ž) .

ℍ𝑁(π‘Ž) = βˆ‘ 1 πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

= βˆ‘ 1

(βˆ’1)𝑛+π‘Ž+1𝐹𝑛 =

𝑁

𝑛=1

βˆ‘(βˆ’1)𝑛+π‘Ž+1 𝐹𝑛

𝑁

𝑛=1

= (βˆ’1)π‘Ž+1βˆ‘(βˆ’1)𝑛 𝐹𝑛

𝑁

𝑛=1

= (βˆ’1)π‘Ž+1𝔾𝑁(π‘Ž).

∎

ALTERNATING SUM

The alternating sum of Fibonacci numbers of negative indices of 1st order is, 𝕀𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛

πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

. We express this in terms of 𝔽𝑁(π‘Ž) .

𝕀𝑁(π‘Ž) = βˆ‘(βˆ’1)𝑛 πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

= βˆ‘ (βˆ’1)𝑛 (βˆ’1)𝑛+π‘Ž+1𝐹𝑛 =

𝑁

𝑛=1

βˆ‘(βˆ’1)π‘Ž+1 𝐹𝑛

𝑁

𝑛=1

= (βˆ’1)π‘Ž+1βˆ‘ 1 𝐹𝑛

𝑁

𝑛=1

= (βˆ’1)π‘Ž+1𝔽𝑁.

∎

ORDER 2

We next consider the following 2nd order sums:

ℍ𝑁(0, π‘Ž) = βˆ‘ 1 πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

,

𝕀𝑁(0, π‘Ž) = βˆ‘ (βˆ’1)𝑛 πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

,

ℍ𝑁(π‘Ž, 𝑏) = βˆ‘ 1 πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘

𝑁

𝑛=1

,

𝕀 𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛 πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘

𝑁

𝑛=1

.

We express the above sums in terms of 𝔽𝑁(0, π‘Ž), 𝔾𝑁(0, π‘Ž), 𝔽𝑁(π‘Ž, 𝑏) and 𝔾𝑁(π‘Ž, 𝑏).

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Page | 21

NON-ALTERNATING SUM

We consider the following 2nd order non-alternating sums.

ℍ𝑁(0, π‘Ž) = βˆ‘ 1 πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

π‘€β„Žπ‘’π‘Ÿπ‘’ π‘Ž > 0,

ℍ𝑁(π‘Ž, 𝑏) = βˆ‘ 1 πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘

𝑁

𝑛=1

π‘€β„Žπ‘’π‘Ÿπ‘’ 0 < π‘Ž < 𝑏.

We express the above sums in terms of 𝔽𝑁(0, π‘Ž) and 𝔽𝑁(π‘Ž, 𝑏) respectively.

First we consider the sum,

ℍ𝑁(π‘Ž) = βˆ‘ 1 πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

= βˆ‘ 1

(βˆ’1)2𝑛+π‘Ž+2𝐹𝑛 =

𝑁

𝑛=1

βˆ‘ (βˆ’1)π‘Ž 𝐹𝑛𝐹𝑛+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Žβˆ‘ 1 𝐹𝑛𝐹𝑛+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Žπ”½π‘(0, π‘Ž).

∎ We next consider,

ℍ𝑁(π‘Ž, 𝑏) = βˆ‘ 1 πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘

𝑁

𝑛=1

= βˆ‘ 1

(βˆ’1)2𝑛+π‘Ž+𝑏+2𝐹𝑛+π‘ŽπΉπ‘›+𝑏

𝑁

𝑛=1

= βˆ‘ (βˆ’1)π‘Ž+𝑏 𝐹𝑛+π‘ŽπΉπ‘›+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Ž+π‘βˆ‘ 1 𝐹𝑛+π‘ŽπΉπ‘›+𝑏

𝑁

𝑛=1

= (βˆ’1)π‘Ž+𝑏𝔽𝑁(π‘Ž, 𝑏).

∎

ALTERNATING SUM

We consider the following 2nd order alternating sums.

𝕀𝑁(0, π‘Ž) = βˆ‘ (βˆ’1)𝑛 πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

π‘€β„Žπ‘’π‘Ÿπ‘’ π‘Ž > 0,

𝕀𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛 πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘

𝑁

𝑛=1

π‘€β„Žπ‘’π‘Ÿπ‘’ 0 < π‘Ž < 𝑏.

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Page | 22 We express the above sums in terms of 𝔾𝑁(0, π‘Ž) and 𝔾𝑁(π‘Ž, 𝑏).

First we consider the sum, 𝕀𝑁(π‘Ž) = βˆ‘ (βˆ’1)𝑛

πΉβˆ’π‘›πΉβˆ’π‘›βˆ’π‘Ž

𝑁

𝑛=1

= βˆ‘ (βˆ’1)𝑛 (βˆ’1)2𝑛+π‘Ž+2𝐹𝑛=

𝑁

𝑛=1

βˆ‘(βˆ’1)𝑛+π‘Ž 𝐹𝑛𝐹𝑛+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Žβˆ‘(βˆ’1)𝑛 𝐹𝑛𝐹𝑛+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Žπ”Ύπ‘(0, π‘Ž).

∎ Next we consider,

𝕀𝑁(π‘Ž, 𝑏) = βˆ‘ (βˆ’1)𝑛 πΉβˆ’π‘›βˆ’π‘ŽπΉβˆ’π‘›βˆ’π‘

𝑁

𝑛=1

= βˆ‘ (βˆ’1)𝑛

(βˆ’1)2𝑛+π‘Ž+𝑏+2𝐹𝑛+π‘ŽπΉπ‘›+𝑏

𝑁

𝑛=1

= βˆ‘(βˆ’1)𝑛+π‘Ž+𝑏 𝐹𝑛+π‘ŽπΉπ‘›+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Ž+π‘βˆ‘ (βˆ’1)𝑛 𝐹𝑛+π‘ŽπΉπ‘›+𝑏

𝑁

𝑛=1

= (βˆ’1)π‘Ž+𝑏𝔾𝑁(π‘Ž, 𝑏).

∎

SUM WITH INDICES IN A.P.

Let β„Ž be a natural number. We convert the following sums in terms of π”½π‘β„Ž and π”Ύπ‘β„Ž. β„π‘β„Ž = βˆ‘ 1

πΉβˆ’β„Žπ‘›

𝑁

𝑛=1

, π•€π‘β„Ž = βˆ‘(βˆ’1)𝑛 πΉβˆ’β„Žπ‘›

𝑁

𝑛=1

.

β„π‘β„Ž = βˆ‘ 1 πΉβˆ’β„Žπ‘›

𝑁

𝑛=1

= βˆ‘ 1

(βˆ’1)β„Žπ‘›+1πΉβ„Žπ‘›

𝑁

𝑛=1

= βˆ‘(βˆ’1)β„Žπ‘›+1 πΉβ„Žπ‘›

𝑁

𝑛=1

= βˆ’ βˆ‘((βˆ’1)β„Ž)𝑛 πΉβ„Žπ‘›

𝑁

𝑛=1

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Page | 23

= {

βˆ’ βˆ‘(βˆ’1)𝑛 πΉβ„Žπ‘›

𝑁

𝑛=1

if β„Ž is odd,

βˆ’ βˆ‘ 1 πΉβ„Žπ‘›

𝑁

𝑛=1

if β„Ž is even.

= {βˆ’π”Ύπ‘β„Ž if β„Ž is odd,

βˆ’π”½π‘β„Ž if β„Ž is even.

∎

π•€π‘β„Ž = βˆ‘(βˆ’1)𝑛 πΉβˆ’β„Žπ‘›

𝑁

𝑛=1

= βˆ‘ (βˆ’1)𝑛 (βˆ’1)β„Žπ‘›+1πΉβ„Žπ‘›

𝑁

𝑛=1

= βˆ‘(βˆ’1)β„Žπ‘›+1+𝑛 πΉβ„Žπ‘›

𝑁

𝑛=1

= βˆ’ βˆ‘((βˆ’1)β„Ž+1)𝑛 πΉβ„Žπ‘›

𝑁

𝑛=1

= {

βˆ’ βˆ‘ 1 πΉβ„Žπ‘›

𝑁

𝑛=1

if β„Ž is odd,

βˆ’ βˆ‘(βˆ’1)𝑛 πΉβ„Žπ‘›

𝑁

𝑛=1

if β„Ž is even.

= {βˆ’π”½π‘β„Ž if β„Ž is odd,

βˆ’π”Ύπ‘β„Ž if β„Ž is even.

∎

NON-ALTERNATING SUM OF ORDER 1

We consider the problem of finding 1st order non-alternating sum of Fibonacci number with negative indices, β„π‘β„Ž(π‘Ž).

β„π‘β„Ž(π‘Ž) = βˆ‘ 1 πΉβˆ’β„Žπ‘›βˆ’π‘Ž

𝑁

𝑛=1

where β„Ž > 0 and β„Ž|π‘Ž.

We write the above sum in terms of 𝔽𝑁(π‘Ž) and 𝐺𝑁(π‘Ž).

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Page | 24 β„π‘β„Ž(π‘Ž) = βˆ‘ 1

πΉβˆ’β„Žπ‘›βˆ’π‘Ž

𝑁

𝑛=1

= βˆ‘ 1

(βˆ’1)β„Žπ‘›+π‘Ž+1πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= βˆ‘(βˆ’1)β„Žπ‘›+π‘Ž+1 πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Ž+1βˆ‘(βˆ’1)β„Žπ‘› πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Ž+1βˆ‘((βˆ’1)β„Ž)𝑛 πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= (βˆ’1)π‘Ž+1 {

βˆ‘(βˆ’1)𝑛 πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

if β„Ž is odd,

βˆ‘ 1

πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

if β„Ž is even.

= {(βˆ’1)π‘Ž+1π”Ύπ‘β„Ž(π‘Ž) if β„Ž is odd, (βˆ’1)π‘Ž+1π”½π‘β„Ž(π‘Ž) if β„Ž is even.

∎

ALTERNATING SUM OF ORDER 1

We consider the problem of finding 1st order non-alternating sum of Fibonacci number with negative indices, π•€π‘β„Ž(π‘Ž).

π•€π‘β„Ž(π‘Ž) = βˆ‘ (βˆ’1)𝑛 πΉβˆ’β„Žπ‘›βˆ’π‘Ž

𝑁

𝑛=1

where β„Ž > 0 and β„Ž|π‘Ž.

We express the above sum in terms of 𝔽𝑁(π‘Ž) and 𝐺𝑁(π‘Ž).

π•€π‘β„Ž(π‘Ž) = βˆ‘ (βˆ’1)𝑛 πΉβˆ’β„Žπ‘›βˆ’π‘Ž

𝑁

𝑛=1

= βˆ‘ (βˆ’1)𝑛

(βˆ’1)β„Žπ‘›+π‘Ž+1πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

= βˆ‘(βˆ’1)β„Žπ‘›+π‘Ž+1 πΉβ„Žπ‘›+π‘Ž

𝑁

𝑛=1

Figure

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