What are some situations where growth occurs arithmetically and what are situations where growth occurs exponentially?
Exponential (or geometric) growth is fairly common, especially when resources are plentiful, but it doesn't last long. What has to happen for exponential growth to occur is that the population has to be pretty far below the carrying capacity of the environment. Births and deaths, immigration and emigration must take place constantly. So -- any time that's true (most animals) and there's plenty of resources, you get exponential growth for a while.
Arithmetic growth is less common. It happens in things like annual grasses and grasshoppers. For this to occur, you need all the births to occur at once and all the deaths to occur at once (but before the births). In the time before the births, individuals can immigrate and emigrate.
In arithmetic progression every next term increases with a fixed quantity:A.P is with first tem a and a common difference d is: a, a+d, a+2d, a+3d, a+4d, etc
The exponentioal growth is also geometric progression, where every next term increases or decreases by a fixed ratio: The exponential initial population a and growth factor x is of the type:
a, a(1+x) , a(1+x)^2, a(1+x)^3, a(1+x)^4, etc
The exponential growth of a population a, with a fixed ratio or factor could be like: a, ax, ax^2, ax^3, ax^4, etc,
Situations: In practical situations the amount of growth of $100 by a simple interest of 5% annually . The amount you are likely to get along with interest if you invest for 1 year , 2 year, 3 year , 4 year etc is: $105, $110, $115, $120, etc...
If you invest an amount of $100 in compound interest of 5% (annual compounding) ,for a period of an year , 2years, 3years, 4 years etc, your amount grows exponentially like.
100*1.05, 100*1.05^2, 100*1.05^3, 100*1.05^4, ...etc.
Also you can see that human population grows exponentially, but with several constraints like food and space of the planet.There are similar exponential growth in living organisms in Biolological situations, where the species increase exponentially and then decrease exponentially due nature's control.
Example: There are 10000 number of paticular species. If the growth rate is 10 % every 6 months how long it take for the population to double:
10000(1+10/100)^n =2*10000, whre n is the number of 6 months required for the population to double. Solution is n = (log2/log1.1 ) of 6 months.
The concept of arithmetic and geometric growth rates was popularised by the economist Robert Thomas Malthus (1766-1834), when he observed that growth of population grows geometrically whereas the food required to feed the growing population can only grow arithmetically.
In arithmetic progression or growth rate the increase in quantity for a given period is fixed. In arithmetic progression are each term in a series of numbers is obtained by adding a fixed number to the previous term. For example the series below is formed by adding 2 to each subsequent term.
0, 2, 4, 6, 8, 10, 12 ...
An example of arithmetic growth is the cumulative output of a machine measured on daily basis. Here the growth is arithmetic because it is dependent on the productive capacity of the machine, which remains constant.
In geometric progression or growth rate each term in a series of numbers is obtained by multiplying the previous term by a fixed number. For example, the series below each term is double the previous term.
1, 2, 4, 8. 16, 32, 64, 128, ...
A classic example of geometric growth is population. This is result of the fact that the growth in population is proportional to cumulative population.