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I was a math major in college but it has been years since I have been deep into the subject matter. When using mathematical modeling in a simple Cartesian graph, what are the main differences between geometric and exponential growth? I knew this at one point long ago but I do not remember.





In geometric growth, the change is discrete, while in exponential growth, it is continuous.
This site has a useful illustration and formulas: http://fig.cox.miami.edu/~schultz/fall98/17res.html 




I don't think so
Geometric growth has a constant rate of change  the increases per time period are constant. Here's an example, with constant delta of 1 1, 2, 3, 4, 5, 6, ... Exponential growth is where the rate of change is itself increasing. In this example, each number is double the previous 1, 2, 4, 8, 16, ... Russell 






An Exponential function is a function of a constant number of exponential terms. It is a continuous curve; the exponents are from the set of real numbers
Exponential growth (one term): Code:
f(x) = 2^{x} f(0) = 2^{0} = 1 f(1) = 2^{1} = 2 f(2) = 2^{2} = 4 f(3) = 2^{3} = 8 f(4) = 2^{4} = 16 f(5) = 2^{5} = 32 f(6) = 2^{6} = 64 f(7) = 2^{7} =128 f(8) = 2^{8} =256 f(9) = 2^{9} =512 Also f(0.5) = 2^{0.5} = 1.41... f(1) = 1/2^{1} = 0.5 Code:
x g(x) = sum 2^{x} i=0 g(0) = 2^{0} = 1 g(1) = 2^{0} + 2^{1} = 3 g(2) = 2^{0} + 2^{1} + 2^{2} = 7 g(3) = 2^{0} + ... + 2^{3} = 15 g(4) = 2^{0} + ... + 2^{4} = 31 g(5) = 2^{0} + ... + 2^{5} = 63 g(6) = 2^{0} + ... + 2^{6} = 127 g(7) = 2^{0} + ... + 2^{7} = 255 g(8) = 2^{0} + ... + 2^{8} = 511 g(9) = 2^{0} + ... + 2^{9} = 1,023 But g(0.5) is undefined g(1) is undefined 




RusselM and CKDextHavn: What you describe as a geometric growth is actually what's referred to as an arithmatic growth. Gilligan got it right: The only difference between geometric and exponential is that the former is discreet, while the latter is continuous: For any geometric progression, you can find an exponential progression that matches it at all points where it is defined.





I'm sure that 13 years ago, he did indeed mean discrete. But we can forgive Chronos this one trespass; he's more than made up for it over the past decade.







As a retired professional mathematician with a PhD in mathematics, I'm probably a reputable source.
A geometric progression (or sequence) is almost the same as exponential growth which is more properly called an exponential progression (or sequence). A geometric progression starts with a number which I will call a and then is followed by numbers based on a number that I will call b as follows: a, a*b, a*b^2,a*b^3,a*b^4 and so on. If it is a finite geometric progression it stops at some number a*b^k where k is a positive integer. If it is an infinite geometric progression it continues forever  meaning that there is no last integer k as there is in the finite example. For those who are interested this is a countably infinite sequence  the first infinity is countable infinity and it is the number of integers. This may seem strange, but the number of rational numbers, numbers which are the ratio of two integers m and n such that (m/n) is an irreducible fraction, although being irreducible is not necessary to what follows, is also countably infinite. In other words, the number of integers is the same as the number of rational numbers. An exponential sequence starts out with a number and continues as follows: a, a^2, a^3, a^4... which again may be a finite sequence ending in some number a^k where k is a positive integer or it may be (countably) infinite, that is it keeps on going forever (see above). Thus a geometric sequence starts with a number a and then is followed by multiplication of a number b as stated above. Since b can be the same as a, all exponential sequences are geometric sequences, but when a and b are different we have a geometric sequence which is not an exponential sequence. Hence the set of all exponential sequences is a proper subset of the set of all geometric sequences. Addendum: The proof that the number of integers is the same as the number of rational numbers is fairly simple to comprehend, but it's too lengthy for this forum. If you google "prove that the number of integers is the same as the number of rational numbers" (no quotation marks), I'm sure that you will find a plethora of proofs. 




It's not that hard. The count of rational numbers is no greater than the count of ordered pairs of integers, since every rational number can be expressed as a distinct ordered pair of integers. And the set of ordered pairs of integers is equal to the set of ordered pairs of integers whose sum is 0, union the set of ordered pairs of integers whose sum is 1, union the set of ordered pairs of integers whose sum is 2, etc. So enumerate all of the ordered pairs of sum 0, all of the ordered pairs of sum 1, all of the ordered pairs of sum 2, and so on in that order.





On the other hand, headbanger's nitpick while hardly discreet, was the essence of discrete  his one and only post on these Boards.





I'm having a little trouble grasping this. The definition of discrete linked to in post #9, and the difference between discrete and continuous as used here, are not really making sense to me. Is it right that a discrete function can be likened to a series of dots that have spaces in between while a continuous one can be likened to a line?
Best I can figure, an exponential progression is a subset of the set of geometric progressions, and is for lack of a better word "pure." Is it right that x^2, x^3, x^4 etc. is geometric and exponential while 2x^2, 2x^3, 2x^4 etc. is geometric but not exponential? I'm trying to keep it very basic, as I've found the explanations already offered to be confusing. I'm just trying to picture examples that convey the gist of it, so simple laymanoriented replies will probably be most helpful. Thanks. 






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