jewish_scientist wrote:This is what I thought:
Then I found
this and am really confused. It says that geometric growth is another form of exponential growth. Is geometric pattern not a synonym for geometric growth, or am I missing something?
I think part of the problem may be that you've got the same definition for "pattern" (what I would call a "sequence") and "growth". "Patterns" (sequences) have inputs defined over natural numbers: first term a
_{0}, second term a
_{1}, third term a
_{2}, and so on. "Growth" is the interpolation of the sequences to values for inputs outside of/between natural numbers, to create functions which describe the value for any possible input.
For instance, take your arithmetic pattern, and let's assume n > 3: T
_{n} = T
_{n-1} + p (I've renamed "something" with "p")
Then T
_{n} = (T
_{n-2} + p) + p = T
_{n-2} + 2*p
And T
_{n} = ((T
_{n-3} + p) + p) + p = T
_{n-3} + 3*p.
In general, T
_{n} = T
_{0} + n*p.
That's just a linear function. Thus, the extension of an arithmetic pattern to a "growth" gives us a linear function, i.e., an arithmetic pattern is a special case of linear growth.
In the same way, a geometric pattern leads to exponential growth:
(Given n > 3, and some common ratio p)
T
_{n} = T
_{n-1}*p
T
_{n} = (T
_{n-1-1}*p)*p = T
_{n-2}*p
^{2}T
_{n} = ((T
_{n-1-1-1}*p)*p)*p = T
_{n-3}*p
^{3}So, T
_{n} = T
_{0}*p
^{n} - but that's just an exponential function (T(x) = T
_{0}*p
^{x}) evaluated for x in the set of natural numbers.
Thus, geometric patterns extend to exponential functions, which we call "exponential growth". Some people will refer to the case where the exponential function is evaluated at discrete points equally spaced as being "geometric growth"; that basically just says "geometric growth" is a special case of exponential growth
I'm not sure what your exponential sequence would lead to. Someone better than I can define that.
Apologies if my explanation seems simplistic or not very well defined.