Some Addition Formulas for Fibonacci, Pell and Jacobsthal Numbers

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.

Keywords

• Fibonacci numbers
• Pell numbers
• Jacobsthal numbers
• 11B39
• 11K31
• 11Y55