1. 2021 Dec 17, Recursion and Geometric Progressions
1. This thought has come to me many times

# 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.