Growth and Decay

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Growth and Decay

Growth and decay refers to a class of problems in mathematics that can be modeled or explained using increasing or decreasing sequences (also called series). A sequence is a series of numbers, or terms, in which each successive term is related to the one before it by precisely the same formula.

There are many practical applications of sequences. One example is predicting the growth of human populations. Population growth or decline has an impact on numerous economic and environmental issues. When the population grows, so does the rate at which waste is produced, which in turn affects growth rate of landfill sites, nuclear waste dumps, and other sources of pollution. Various other growth rates also affect human lives. For instance, the growth rates of investments and savings accounts, affect a person’s economic well-being. Understanding the mathematics of growth is very important. For example, predicting the rate at which renewable resources, including the forests, marine life, and wildlife, naturally replenish themselves, helps prevent excessive harvesting that can lead to population declines and even extinction.

Arithmetic growth and decay

Arithmetic growth is modeled by an arithmetic sequence. In an arithmetic sequence, each successive term is obtained by adding a fixed quantity to the previous term. For example, an investment that earns simple interest (not compounded) increases by a fixed percentage of the principal (original amount invested) in each period that interest is paid. A onetime investment of $1,000, in an account that pays 5% simple interest per year, will increase by $50 per year. The growth of such an investment, left in place for a 10-year period, is given by the sequence, where the first entry corresponds to the balance at the beginning of the first year, the second entry corresponds to the balance at the beginning of the second year and so on. A sequence that models growth is an increasing sequence, one that models decay is a decreasing sequence. For instance, some banks require depositors to maintain a minimum balance in their checking accounts, or else pay a monthly service charge on the account. If an account, with a required minimum of $500, has $50 in it, and the owner stops using the account without closing it, then the balance will decrease arithmetically each month, by the amount of the monthly service charge, until it reaches zero.

Geometric 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, …, Pn = cPn-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

KEY TERMS

Limit— A limit is a bound. When the terms of a sequence that are very far out in the series grow ever closer to a specific finite value, without ever quite reaching it, that value is called the limit of the sequence.

Mathematical model— A mathematical model is the expression of a physical law in terms of a specific mathematical concept.

Rate— A rate is a comparison of the change in one quantity with the simultaneous change in another, where the comparison is made in the form of a ratio.

Sequence— A sequence is a series of terms, in which each successive term is related to the one before it by a fixed formula.

of radioactive decay. When the nucleus of a radioactive element decays, it emits one or more alpha, beta or gamma particles, and becomes stable (non-radioactive). 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 l 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 N₀λt nuclei will decay in the time period t. At the end of the first time period, there will be N₁ = (N0 - N0λt) 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 = cⁿ T₀, 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 ex, 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. (The Fibonacci sequence was named after Italian mathematician Leonardo of Pisa [c. 1170–1250], who was also known as Leonardo Fibonacci, or simply Fibonaccci.) The population growth of male honeybees is an example of a growth rate that follows the Fibonacci sequence.

See also Fibonacci sequence.