Chapter 2: Integral Calculus
Society for Industrial and Applied Mathematics eBooks · 2023
2.1 ▪ Sequences Suppose that the population of a species of insects grows 25% per week. Taking into account that the initial population is 1000, what will be the population size after n weeks? After one week we will have 1000 + 0.25(1000) = (1.25)1000. After two weeks we will have (1.25)1000 + 0.25(1.25)1000 = (1 + 0.25)1.25(1000) = 1.252(1000) and so on. Thus, after n weeks we will have 1.25n(1000). For instance, if n = 13 we will have 1.2513(1000) = 18,190 insects. In general, if we start with a population Po and Pn denotes the population after n weeks, the same reasoning employed before leads to Pn=(1+r)nPo, (2.1) where r is the fixed percentage of growth. Under these circumstances we say that the growth is Malthusian, in honor of the British scholar Thomas R. Malthus (1766-1834). Of course, n can denote weeks, days, or hours, depending on the nature of the problem. Bacteria in a culture with unlimited supply of nutrients and plenty of space often follow Malthusian growth. Mathematically speaking, we have a correspondence between n and Pn, which is usually denoted by (Pn) and is called a sequence. In general, a sequence is any function a : N → ℜ. It is commonly denoted by (an), where an = a(n), or simply a1, a2, a3,… when it is easy to find the value of an. For instance 1, 1/2, 1/3, … denotes the sequence (1/n).