Generalization of Arithmetico-Geometric Series and the Expectation Value of a $k$-Run of a Bernoulli Trial

TL;DR

This paper presents a novel derivation of the expected number of Bernoulli trials needed to observe k consecutive successes, using an arithmetic-geometric Fibonacci series and recurrence relations.

Contribution

The paper introduces a new derivation method for a known result, employing an arithmetic-geometric Fibonacci series and recurrence relations.

Findings

Derivation of the expected value using Fibonacci series

Recurrence relation approach to the problem

Confirmation of known results through a new method

Abstract

The article uses an arithmetic-geometric Fibonacci series to find the expected value of trials needed to observe k consecutive successes for the first time in a Bernoulli experiment using a recurrence relation. It is important to note that this is not a new result, but to the best of the authors' knowledge at the time of uploading, this is a novel derivation of a well-established result. The other derivations of this result are cited in the references section.