Growth and Decay

Geometric growth and decay are modeled with geometric sequences. A geometric sequence is one in which each successive term is multiplied by a fixed quantity. In general, a geometric sequence is one of the form, where P₁ = cP₀, P₂ = cP₁, P₃ = cP₂,..., P_n = cP_n-1, and c is a constant called the common ratio. If c is greater than 1, the sequence is increasing. If c is less than 1, the sequence is decreasing. The rate at which an investment grows when it is deposited in an account that pays compound interest is an example of a geometric growth rate. Suppose an initial deposit of P₀ is made in a bank paying a fixed interest rate that is compounded annually. Let the interest rate in decimal form be r. Then, the account balance at the end of the first year will be P₁ = (P₀ + r P₀) = (1+r) P₀. At the end of the second year, the account balance will be P₂ = (P₁ + rP₁) = (1+r)P₁. By continuing in this way it is easy to see that the account balance in any given year will be equal to (1+r) times the previous years balance. Thus, the growth rate of an initial investment earning compound interest is given by the geometric sequence that begins with the initial investment, and has a common ratio equal to the interest rate plus 1.

This same compounding model can be applied to population growth. However, unlike the growth of an investment, population growth is limited by the availability of food, water, shelter, and the prevalence of disease. Thus, population models usually include a variable growth rate, rather than a fixed growth rate, that can take on negative as well as positive values. When the growth rate is negative, a declining population is predicted. One such model of population growth is called the logistic model. It includes a variable growth rate that is obtained by comparing the population in a given year to the capacity of the environment to support a further increase. In this model, when the current population exceeds the capacity of the environment to support the population, the quantity in parentheses becomes negative, causing a subsequent decline in population.

Still another example of a process that can be modeled using a geometric sequence is the process of radioactive decay. When the nucleus of a radioactive element decays it emits one or more alpha, beta or gamma particles, and becomes stable (nonradioactive). This decay process is characteristic of the particular element undergoing decay, and depends only on time. Thus, the probability that one nucleus will decay is given by: Probability of Decay = λt, where λ depends on the element under consideration, and t is an arbitrary, but finite (not infinitesimally short), length of time. If there are initially N₀ radioactive nuclei present, then it is probable that N₀ λt nuclei will decay in the time period t. At the end of the first time period, there will be N₁ = (N₀ - N₀ λt) or N₁ = N₀ (1- λt) nuclei present. At the end of the second time period, there will be N₂ = N₁ (1- λt), and so on. Carrying this procedure out for n time periods results in a sequence similar to the one describing compound interest, however, λ is such that this sequence is decreasing rather than increasing. In order to express the number of radioactive nuclei as a continuous function of time rather than a sequence of separated times, it is only necessary to recognize that t must be chosen infinitesimally small, which implies that the number of terms, n, in the sequence must become infinitely large. To accomplish this, the common ratio is written (1 - λt/n), where t/n will become infinitesimally small as n becomes infinitely large. Since a geometric sequence has a common ratio, any term can be written in the form T_n+1 = cⁿT₀, where T₀ is the initial term, so that the number of radioactive nuclei at any time, t, is given by the sequence N = N₀(1 - λt/n)ⁿ when n approaches infinity. It is well known that the limit of the expression (1 + x/n)ⁿ as n approaches infinity equals e^x, where e is the base of the natural logarithms. Thus, the number of radioactive nuclei present at any time, t, is given by N = N₀e^λt , where N₀ is the number present at the time taken to be t = 0.

Finally, not all growth rates are successfully modeled by using arithmetic or geometric sequences. Many growth rates are patterned after other types of sequences, such as the Fibonacci sequence, which begins with two 1s, each term thereafter being the sum of the two previous terms. Thus, the Fibonacci sequence is. The population growth of male honeybees is an example of a growth rate that follows the Fibonacci sequence.