<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

Published Online: Jul 18, 2019

Page range: 55 - 65

Received: Nov 21, 2018

Accepted: May 01, 2019

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

Keywords
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

Published Online: Jul 18, 2019

Page range: 55 - 65

Received: Nov 21, 2018

Accepted: May 01, 2019

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

KeywordsFibonacci 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.

Keywords
Fibonacci numbers, Pell numbers, Jacobsthal numbers