Geometric Growth vs. Geometric Pattern

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jewish_scientist
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Geometric Growth vs. Geometric Pattern

Postby jewish_scientist » Fri Dec 09, 2016 4:30 pm UTC

This is what I thought:
Spoiler:
Arithmetic Pattern: Tn = Tn-1 + something
Geometric Pattern: Tn = Tn-1 * something
Exponential Pattern: Tn = Tn-1 ^ something

Arithmetic Growth: Tn = Tn-1 + something
Geometric Growth: Tn = Tn-1 * something
Exponential Growth: Tn = Tn-1 ^ something


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?

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Flumble
Yes Man
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Re: Geometric Growth vs. Geometric Pattern

Postby Flumble » Fri Dec 09, 2016 6:37 pm UTC

Ah, no, exponential growth is merely T_n = T_{n-1} * something.

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WibblyWobbly
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Re: Geometric Growth vs. Geometric Pattern

Postby WibblyWobbly » Fri Dec 09, 2016 6:47 pm UTC

jewish_scientist wrote:This is what I thought:
Spoiler:
Arithmetic Pattern: Tn = Tn-1 + something
Geometric Pattern: Tn = Tn-1 * something
Exponential Pattern: Tn = Tn-1 ^ something

Arithmetic Growth: Tn = Tn-1 + something
Geometric Growth: Tn = Tn-1 * something
Exponential Growth: Tn = Tn-1 ^ something


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 a0, second term a1, third term a2, 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: Tn = Tn-1 + p (I've renamed "something" with "p")
Then Tn = (Tn-2 + p) + p = Tn-2 + 2*p
And Tn = ((Tn-3 + p) + p) + p = Tn-3 + 3*p.
In general, Tn = T0 + 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)
Tn = Tn-1*p
Tn = (Tn-1-1*p)*p = Tn-2*p2
Tn = ((Tn-1-1-1*p)*p)*p = Tn-3*p3
So, Tn = T0*pn - but that's just an exponential function (T(x) = T0*px) 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.

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Eebster the Great
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Re: Geometric Growth vs. Geometric Pattern

Postby Eebster the Great » Sat Dec 10, 2016 5:49 am UTC

The third type of sequence is superexponential, yielding the explicit formula Tn = (T0)pⁿ. You get this sequence for p = 2, T0 = 2:
2, 4, 16, 256, 65536, 4294967296, ...


A more interesting variant instead has Tn = pTₙ₋₁. This yields the explicit formula Tn = (ⁿp)T₀, where np is p tetrated to the n. That is, Tn = p^p^...^p^T0 with n copies of p.

That would be "tetrational" growth. You get this sequence for p = 2, T0 = 1:
1, 2, 4, 16, 65536, ...
The next term is about 2.004 × 1019728.

Edit: parentheses added for clarity


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