SO I have a math problem that I forgot how to do.
I need to develop a mathematical model describing the number of cicadas in a given year.
Year: | 1953 | 1970 | 1987 | 2004
Cicadas: | 20 | 38 | 82 | 127
There are many ways to develop such a model. For example, one may try to have a linear function and to minimize the square of error for each year. I did this in OpenOffice Calc using LINEST function and got the following line equation: $\displaystyle y = 2.147x - 4181.2$. See the graphs here: .
With more complex models, in fact, any number can be a prediction, say, for 2011. Indeed, given $\displaystyle n$ points $\displaystyle (x_i,y_i)$, $\displaystyle i=1,\ldots,n$, there exists a Lagrange polynomial $\displaystyle f(x)$ of degree $\displaystyle n - 1$ such that $\displaystyle f(x_i)=y_i$ for all $\displaystyle i=1,\dots,n$.
Well, the number of cicadas went from 20 to 38 (which is almost 40) in 17 years, then went from 38 to 82, which just slightly larger than 80 in 17 years, but in the final 17 years only increased by maybe 3/2 rather than doubling.
Letting N(t) be the number of cicadas t years after 1953 suggests $\displaystyle 20(2^{t/17}$ for the first three data points but the last one does not fit.