<div dir="ltr"><div dir="ltr"><br></div><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><br>---------- Forwarded message ----------<br>From: Rick Moen <<a href="mailto:rick@linuxmafia.com" target="_blank">rick@linuxmafia.com</a>><br>To: <a href="mailto:conspire@linuxmafia.com" target="_blank">conspire@linuxmafia.com</a><br>Cc: <br>Bcc: <br>Date: Tue, 24 Mar 2020 20:36:35 -0700<br>Subject: Re: [conspire] Correction<br>Quoting Texx (<a href="mailto:texxgadget@gmail.com" target="_blank">texxgadget@gmail.com</a>):<br>
<br>
> Yesterday, you rattled off a numerical progression based on doubles.<br>
> To me that is geometric.<br>
<br>
No, it really isn't.<br>
<br>
Please allow the guy with the math degree to disambiguate. Quoting<br>
Wikipedia (in part) to save time:<br>
<br>
A _geometric_ progression is one where each term after the first is<br>
found by multiplying the previous one by a fixed, non-zero number called<br>
the common ratio. For example, the sequence 2, 6, 18, 54,... is a<br>
geometric progression with common ratio 3.<br>
<br></blockquote><div><br></div><div>6/2=18/6=54/18 = 3</div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
An _exponential_ progression is one where each term after the first <br>
is the prior term multiplied by a factor with an exponent whose value<br>
(the exponent's value) goes up by one more with each term. For example, <br>
the sequence 200, 400, 800, 1600, 3200 is an exponential progression<br></blockquote><div><br></div><div>400/200=800/400=1600/800=3200/1600 = 2</div><div><br></div><div><br></div><div>Both examples are geometric sequence. </div><div><br></div><div>A geometric sequence is one where the ratio of each to the next is the same. </div><div><br></div><div>S(n+1)/S(n) is a constant. </div><div>If you call that common ratio r and start your numbering at 0 then, </div><div><br></div><div>S(n)= S(0)*r^n</div><div><br></div><div>Now an exponential curve is one where the rate of change is proportional to itself. </div><div><br></div><div>If it is exponential growth then it is growing even faster the bigger it gets. </div><div>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. </div><div><br></div><div>"the rate of change is proportional to itself" </div><div>That's a differential equation: </div><div><br></div><div>dx/dt=k*x(t) where k is a constant. </div><div><br></div><div>The solution of that is the exponential function: </div><div><br></div><div>x(t)= x(0)*e^(k*t) where e is the base of the natural logarithm. </div><div>Same: </div><div>x(t)=X(0)*(e^k)^t </div><div><br></div><div>If you let K be e^k then you get: </div><div><br></div><div>x(t)=x(0)*K^t ---The same formula we get for the geometric sequence "S(n)= S(0)*r^n" </div><div><br></div><div>So exponential and geometric are pretty much the same thing: one is continuous and the other is discrete.</div><div><br></div><div>{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. </div><div><br></div><div>I was a Math guy too in a previous life and it was just bothering me, </div><div><br></div><div>Thomas</div><div><br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
with multiplication factor 2^(t-1) between terms, where t is the term<br>
index number.<br>
<br>
The point is that exponential sequences curve upwards very dramatically <br>
after a few terms, in exactly the way geometric ones (let alone linear<br>
ones) do not.<br>
<br>
Most laymens' habits of thinking quickly revert to imagining linear<br>
progressions, because that's mostly what they encounter from day to day,<br>
_even_ when the explainer is very specific about the matter under<br>
discussion involving exponential growth -- as with the algae on the pond.<br>
<br>
<br><br>
</blockquote></div></div>