In this book we consider the concept of sequences in mathematics. We give a brief discussion of some know concepts and theorems like "Sarkoviskii's Theorem." Recently, many authors consider new concepts like rational sequences and piece-wise defined sequences. We talk about this new trend and develop our own concepts. We prove many results regarding these new concepts. Specifically, we focus on the sequences defined as Q_=hQ_{k-1}, if Q_{k-1}v, for k=1,2, ... . where h, j, v, and Q₀ are fixed real numbers. We and J. Guyker pubilshed a paper entitled "Nonlinear Piecewise De...fined...