Abstract
Abstract In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known recursive sequences, as the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, etc. We establish some general limit formulas, where the product of the first n terms of these sequences appears. Furthermore, we prove some general limits that connect these sequences to the number e(≈ 2:71828:::).
Highlights
A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given
Further examples of sequences in recursive forms (1.1) and (1.2) are the Pell and Pell–Lucas sequences that the first is of the recursive form (1.1) and the second is of the recursive form (1.2)
We prove some general limit formulas, where the product of the first n terms of sequences (1.1) and (1.2) appears
Summary
A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given. Sequences (1.1) and (1.2) can be defined by the following closed-form solutions (known as Binet style formulas), respectively:. Further examples of sequences in recursive forms (1.1) and (1.2) are the Pell and Pell–Lucas sequences that the first is of the recursive form (1.1) and the second is of the recursive form (1.2). These sequences are defined as follows (for n ≥ 0), respectively: Pn+2 = 2Pn+1 + Pn, P0 = 0, P1 = 1, Qn+2 = 2Qn+1 + Qn, Q0 = 2, Q1 = 2. The following closed-form solutions (according to (1.3) and (1.4)) exist for the Pell and Pell–Lucas numbers, respectively:. We prove some limit formulas that connect sequences (1.1) and (1.2) to the number e(≈ 2.71828...)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
