Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

Abstract

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

Notes on a General Sequence

“…The Lucas, Jacobsthal, Pell, Pell-Lucas, and Jacobsthal-Lucas numbers are respectively. For more information see, [2], [4], [6], and [9]. Similarly, the following well-known formulae exist for the Lucas, Jacobsthal, Pell, Pell-Lucas, and Jacobsthal-Lucas numbers (see [5]), respectively:…”