Delving Deeper: Growth Rates and the Marvelous Geometric Sequence

Abstract

Students are often dazzled by the prodigious growth rate of the geometric sequence gn = 2n and the geometric series whose partial sums are Sn = 1 + 2 + 4 + 8 + … + 2n−1 = 2n − 1. Teachers sometimes note that the geometric sequence is the discrete “form” of an exponential function, which is characterized by very rapid growth. In particular, exponential functions grow faster than polynomial functions. A rigorous explanation of this claim is left to the calculus class in which students examine the relative growth rates of functions by using L'Hopital's rule. However, even by using tools developed in algebra and precalculus, teachers can explain to their students (or lead them to a justification) that the geometric sequence gn = 2n grows faster than any polynomial sequence. These tools are the binomial theorem, Pascal's triangle, mathematical induction, and an understanding of the end behavior of exponential and polynomial functions (i.e., what happens to the graphs as x approaches infinity).