Fibonacci Partial Sums Tricks
Fuente:
arXiv
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| Autori principali: | , , , , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916379327725568 |
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| author | Byrapuram, Nikhil Ge, Adam Ge, Selena Khovanova, Tanya Lee, Sylvia Zia Mandal, Rajarshi Redwine, Gordon Samanta, Soham Wu, Daniel Xu, Danyang Zhao, Ray |
| author_facet | Byrapuram, Nikhil Ge, Adam Ge, Selena Khovanova, Tanya Lee, Sylvia Zia Mandal, Rajarshi Redwine, Gordon Samanta, Soham Wu, Daniel Xu, Danyang Zhao, Ray |
| contents | The following magic trick is at the center of this paper. While the audience writes the first ten terms of a Fibonacci-like sequence (the sequence following the same recursion as the Fibonacci sequence), the magician calculates the sum of these ten terms very fast by multiplying the 7th term by 11. This trick is based on the divisibility properties of partial sums of Fibonacci-like sequences. We find the maximum Fibonacci number that divides the sum of the Fibonacci numbers 1 through $n$. We discuss the generalization of the trick for other second-order recurrences. We show that a similar trick exists for Pell-like sequences and does not exist for Jacobhstal-like sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01296 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fibonacci Partial Sums Tricks Byrapuram, Nikhil Ge, Adam Ge, Selena Khovanova, Tanya Lee, Sylvia Zia Mandal, Rajarshi Redwine, Gordon Samanta, Soham Wu, Daniel Xu, Danyang Zhao, Ray History and Overview Number Theory 11B39 (Primary) 00A08 The following magic trick is at the center of this paper. While the audience writes the first ten terms of a Fibonacci-like sequence (the sequence following the same recursion as the Fibonacci sequence), the magician calculates the sum of these ten terms very fast by multiplying the 7th term by 11. This trick is based on the divisibility properties of partial sums of Fibonacci-like sequences. We find the maximum Fibonacci number that divides the sum of the Fibonacci numbers 1 through $n$. We discuss the generalization of the trick for other second-order recurrences. We show that a similar trick exists for Pell-like sequences and does not exist for Jacobhstal-like sequences. |
| title | Fibonacci Partial Sums Tricks |
| topic | History and Overview Number Theory 11B39 (Primary) 00A08 |
| url | https://arxiv.org/abs/2409.01296 |