Choosing Between Two Competing Models

Two models can start out looking like twins. A line and an exponential curve can both pass through the same first two points, since two points determine a unique line and also pin down a unique exponential through them. But a line grows by adding the same amount every step, while an exponential grows by multiplying by the same factor every step. Addition and multiplication pull apart fast: adding 4 repeatedly gives 4, 8, 12, 16; multiplying by 4 repeatedly gives 4, 16, 64, 256. Early on the gap is small. By the fourth or fifth step it is enormous.

The test for which family fits is mechanical and reliable. Take the outputs at evenly spaced inputs and look at neighboring values. Subtract each pair: if those differences are all the same number, the data is linear, and that constant difference is the slope. Instead divide each pair: if those ratios are all the same number, the data is exponential, and that constant ratio is the base. You only need one of the two tests to come out constant to identify the family — differences constant means linear, ratios constant means exponential, and real data rarely gives you both.

This is the same split that separates arithmetic sequences from geometric sequences: arithmetic sequences add a common difference each term, geometric sequences multiply by a common ratio. Choosing a model family for a graph or a table is exactly choosing which kind of sequence its outputs form. Once you know the family, writing the rule is just plugging the starting value and the constant difference or ratio into the linear or exponential form.