Generalized Fibonacci Numbers: Sum Formulas
Yüksel Soykan *
Department of Mathematics, Art and Science Faculty, Zonguldak Bülent Ecevit University, 67100 Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this paper, closed forms of the summation formulas for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery.
Keywords: Fibonacci numbers, Lucas numbers, Pell numbers, Jacobsthal numbers, sum formulas.