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Articles

Vol. 10 No. 2 (2022)

Deriving a Formula in Solving Reverse Fibonacci Means

DOI:
https://doi.org/10.32871/rmrj2210.02.03
Submitted
March 22, 2022
Published
December 30, 2022
FULL PAPER

Keywords

  • fibonacci sequence
  • reverse fibonacci sequence
  • Binet’s formula
  • means

Abstract

Reverse Fibonacci sequence $\{J_n\}$ is defined by the relation $J_n = 8(J_{n-1} - J_{n-2})$ for $n\geq2$ with $J_0=0$ and $J_1=1$ as initial terms. A few formulas have been derived for solving the missing terms of a sequence in books and mathematical journals, but not for the reverse Fibonacci sequence. Thus, this paper derived a formula that deductively solves the first missing term $\{x_1\}$?of the reverse Fibonacci sequence and is given by the equation

$x_1=\frac{b+8aJ_n}{J_{n+1}}$.

By using the derived formula for $\{x_1\}$, it is now possible to solve the means of the reverse Fibonacci sequence as well as solving the sequence itself.

References

  1. Janičko, O. (2018). New fundamental discovery of the Reverse Fibonacci sequence. Academia. Retrieved February 05, 2019, from https://www.academia.edu/38243263/New_fundamental_discovery_of_the_reverse_Fibonacci_sequence#:~:text=The%20reverse%20Fibonacci%20ratio%20is,discovery%20will%20be%20seriously%20recognized
  2. Natividad, L. R. (2011a). Deriving a formula in solving Fibonacci-like sequence. International Journal of Mathematical and Scientific Computing,1(1),19-21. https://veltech.edu.in/wp-content/uploads/2016/04/IJMSC01-02-Paper04.pdf
  3. Natividad, L. R. (2011b). On solving Pell means. International Journal of Mathematical Archive, 2(12), 2736-2739. http://ijma.info/index.php/ijma/article/view/807
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  7. Rabago, J. F. T. (2012). On Natividad’s formula for solving the missing terms of a recurrence sequence. International Journal of Mathematical Archive, 3(8), 3105-3107. https://www.researchgate.net/publication/269371138_On_Natividad'd_Formula_for_solving_the_missing_terms_of_a_recurrence_sequence/citations
  8. Sisodiya, K. S., Singh, B., & Sisodiya, K. (2014). On Lucas sequence formula for solving the missing terms of a recurrence sequence. International Journal of Technology Enhancements and Emerging Engineering Research, 2(5), 142-144. https://issuu.com/ijteee/docs/on-lucas-sequence-formula-for-solvi
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