<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

Publié en ligne: 18 juil. 2019

Pages: 55 - 65

Reçu: 21 nov. 2018

Accepté: 01 mai 2019

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

Mots clés
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

Publié en ligne: 18 juil. 2019

Pages: 55 - 65

Reçu: 21 nov. 2018

Accepté: 01 mai 2019

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

Mots clésFibonacci 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.

Mots clés
Fibonacci numbers, Pell numbers, Jacobsthal numbers