标题： 算术-几何级数的推广及伯努利试验中$k$连续出现的期望值 显示英文标题

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

摘要： 文章使用算术-几何斐波那契级数，通过递推关系找到在伯努利实验中首次观察到k次连续成功所需的试验的期望值。 需要注意的是，这不是一个新结果，但根据作者在上传时的知识，在此之前这是对一个已建立结果的新推导。 该结果的其他推导在参考文献部分引用。 显示英文摘要

摘要： 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.