[conspire] Correction

[paulz at ieee.org]

I don't claim to be a "math guy", but I did take 11 semesters math in college.

Just had to work out the examples geometric and exponential formulas using a LibreOffice spread sheet.

One was "geometric":  multiply each number by 3 to get the next.  2,6,18 ...

Other followed rule for exponential.  At each point calculate the slope.  Then add slope to present value to get the next point.  Golly gee, the answers are the same:


geo             x        dx/dt
2                 2           4
6                 6          12
18              18        36
54              54       108
162          162       324
486          486       972

One of the things I find fascinating about math is that often there are many valid ways to solve the same problem.   Unfortunately, most school curricula only teach one method and if a student happens to use a method not in the text book they are criticized. 






On Thursday, March 26, 2020, 2:47:08 AM PDT, tom r lopes <tomrlopes at gmail.com> wrote: 






> 
> ---------- Forwarded message ----------
> From: Rick Moen <rick at linuxmafia.com>
> To: conspire at linuxmafia.com
> Cc: 
> Bcc: 
> Date: Tue, 24 Mar 2020 20:36:35 -0700
> Subject: Re: [conspire] Correction
> Quoting Texx (texxgadget at gmail.com):
> 
>> Yesterday, you rattled off a numerical progression based on doubles.
>> To me that is geometric.
> 
> No, it really isn't.
> 
> Please allow the guy with the math degree to disambiguate.  Quoting
> Wikipedia (in part) to save time:
> 
> A _geometric_ progression is one where each term after the first is
> found by multiplying the previous one by a fixed, non-zero number called
> the common ratio.  For example, the sequence 2, 6, 18, 54,... is a
> geometric progression with common ratio 3.
> 

6/2=18/6=54/18 = 3
 
>  An _exponential_ progression is one where each term after the first 
> is the prior term multiplied by a factor with an exponent whose value
> (the exponent's value) goes up by one more with each term.  For example, 
> the sequence 200, 400, 800, 1600, 3200 is an exponential progression

400/200=800/400=1600/800=3200/1600 = 2


Both examples are geometric sequence.  

A  geometric sequence is one where the ratio of each to the next is the same.  

S(n+1)/S(n) is a constant.  
If you call that common ratio r and start your numbering at 0 then, 

S(n)= S(0)*r^n

Now an exponential curve is one where the rate of change is proportional to itself.  

If it is exponential growth then it is growing even faster the bigger it gets.  
Even more than that because the speed is also exponential, and the rate of change of the speed (i.e. acceleration) is also exponential. And so on.  

"the rate of change is proportional to itself" 
That's a differential equation:  

dx/dt=k*x(t) where k is a constant.  

The solution of that is the exponential function:  

x(t)= x(0)*e^(k*t)  where e is the base of the natural logarithm.  
Same: 
x(t)=X(0)*(e^k)^t  

If you let K be e^k then you get: 

x(t)=x(0)*K^t  ---The same formula we get for the geometric sequence "S(n)= S(0)*r^n" 

So exponential and geometric are pretty much the same thing:  one is continuous and the other is discrete.

{And if you look at finite differences of sequences you can see that geometric sequences satisfy a difference equation just like the differential equation of the exponential.  

I was a Math guy too in a previous life and it was just bothering me, 

Thomas

>  with multiplication factor 2^(t-1) between terms, where t is the term
> index number.
> 
> The point is that exponential sequences curve upwards very dramatically 
> after a few terms, in exactly the way geometric ones (let alone linear
> ones) do not.
> 
> Most laymens' habits of thinking quickly revert to imagining linear
> progressions, because that's mostly what they encounter from day to day,
> _even_ when the explainer is very specific about the matter under
> discussion involving exponential growth -- as with the algae on the pond.
> 
> 
> 
> 
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