Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

Abstract

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.