Thoughts and Analogies: Recursion and G.P. and finite processes
Table of Contents
2021 Dec 17, Recursion and Geometric Progressions
This thought has come to me many times
Cleaning analogy
When cleaning up something, like a room, we use objects which themselves will get dirty. So, soon or later, we must clean the cleaning objects themselves. In turn, we use other cleaning means, or objects, like clean water. This water, soon or later, will be treated, so to be cleaned. Etc.
The same can be said about a maid, for example. We use money to pay someone to clean our house for us. This person, in turn, will have to pay another person to clean their own, or clean themselves.
Mapping the analogy to simple mathematics
Finally, this structure of things maps to the structure of Geometric Progressions with decreasing terms. That is, the geometric ratio
being less than 1. Series with geometric ratios less than one will be
converging series. The sum total value of the infinite series given
by:
Let \(\mathbf = (a_0, a_1, a_2, \ldots)\), be terms of the
series. Let \(r\) be the ratio between two consecutive terms,
\(r=\dfrac{a_{j+1}}\). Thus, the infinite sum is given by
\(S_{\infty} = \dfrac{1-r}\).
Case example, back to our analogy
So, let's say, in our analogy, that \(\mathbf\) signifies the
consecutive efforts of each task. e.i., \(a_0\) is the effort to do the
first cleaning task. Therefore, if we simplify our ratio to a decreasing
average \(|r|<1\), we have that the \(S_{\infty}\) can be applied.
Thus, let \(a_0=1\) and \(r=0.5\), the total effort to clean, in the entire chain of process is: \(S_{\infty} = \dfrac{1}{1-0.5} = 2\). So, the effort of all cleaning chain is twice the cleaning effort of the first cleaning.
Conclusion
Sometimes we just urge to write down our thoughts, doesn't matter how simplistic or funny they may sound. That's one of those.
Although, entire economic and social models can be arrive from these banal thoughts, with infinitely minused mathematical descriptions.