Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

Abstract

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