<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper, we obtain a closed form for <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0005_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>F</mi></mrow><mrow><msubsup><mo>?</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mrow></mrow></mrow></msub></mrow></math><tex-math>${F_{\sum\nolimits_{i = 1}^k {} }}$</tex-math></alternatives></inline-formula>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0005_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>P</mi></mrow><mrow><msubsup><mo>?</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mrow></mrow></mrow></msub></mrow></math><tex-math>${P_{\sum\nolimits_{i = 1}^k {} }}$</tex-math></alternatives></inline-formula>and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0005_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>J</mi></mrow><mrow><msubsup><mo>?</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mrow></mrow></mrow></msub></mrow></math><tex-math>${J_{\sum\nolimits_{i = 1}^k {} }}$</tex-math></alternatives></inline-formula> for some positive integers <italic>k</italic> where <italic>Fr</italic>, <italic>Pr</italic> and <italic>Jr</italic> are the <italic>r</italic>th Fibonacci, Pell and Jacobsthal numbers, respectively. We also give three open problems for the general cases <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0005_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>F</mi></mrow><mrow><msubsup><mo>?</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow></mrow></mrow></msub></mrow></math><tex-math>${F_{\sum\nolimits_{i = 1}^n {} }}$</tex-math></alternatives></inline-formula>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0005_eq_005.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>P</mi></mrow><mrow><msubsup><mo>?</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow></mrow></mrow></msub></mrow></math><tex-math>${P_{\sum\nolimits_{i = 1}^n {} }}$</tex-math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0005_eq_006.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>J</mi></mrow><mrow><msubsup><mo>?</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow></mrow></mrow></msub></mrow></math><tex-math>${J_{\sum\nolimits_{i = 1}^n {} }}$</tex-math></alternatives></inline-formula>for any arbitrary positive integer <italic>n</italic>.</p></abstract>

In this paper, we obtain a closed form for F?i=1k${F_{\sum\nolimits_{i = 1}^k {} }}$, P?i=1k${P_{\sum\nolimits_{i = 1}^k {} }}$and J?i=1k${J_{\sum\nolimits_{i = 1}^k {} }}$ for some positive integers k where Fr, Pr and Jr are the rth Fibonacci, Pell and Jacobsthal numbers, respectively. We also give three open problems for the general cases F?i=1n${F_{\sum\nolimits_{i = 1}^n {} }}$, P?i=1n${P_{\sum\nolimits_{i = 1}^n {} }}$ and J?i=1n${J_{\sum\nolimits_{i = 1}^n {} }}$for any arbitrary positive integer n.

Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

Department of Computer Education and Instructional Technologies

Department of Mathematics Institute of Science and Technology

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers"}

In this paper, we obtain a closed form for F?i=1k${F_{\sum\nolimits_{i = 1}^k {} }}$, P?i=1k${P_{\sum\nolimits_{i = 1}^k {} }}$and J?i=1k${J_{\sum\nolimits_{i...

Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

Publicado en línea: 18 jul 2019

Páginas: 55 - 65

Recibido: 21 nov 2018

Aceptado: 01 may 2019

DOI: https://doi.org/10.2478/amsil-2019-0005

Palabras clave
Fibonacci numbers, Pell numbers, Jacobsthal numbers

© 2019 Göksal Bilgici et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

Publicado en línea: 18 jul 2019

Páginas: 55 - 65

Recibido: 21 nov 2018

Aceptado: 01 may 2019

DOI: https://doi.org/10.2478/amsil-2019-0005

Palabras claveFibonacci numbers, Pell numbers, Jacobsthal numbers

© 2019 Göksal Bilgici et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Palabras clave
Fibonacci numbers, Pell numbers, Jacobsthal numbers