Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

Abstract

Abstract In this paper, we obtain a closed form for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>k</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${F_{\sum\nolimits_{i = 1}^k {} }}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>k</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${P_{\sum\nolimits_{i = 1}^k {} }}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>k</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${J_{\sum\nolimits_{i = 1}^k {} }}$ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>F</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${F_{\sum\nolimits_{i = 1}^n {} }}$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${P_{\sum\nolimits_{i = 1}^n {} }}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:msubsup> <m:mo>?</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow/> </m:mrow> </m:msub> </m:mrow> </m:math> ${J_{\sum\nolimits_{i = 1}^n {} }}$ for any arbitrary positive integer n.