Suppose I have two series $$s_1 = (1,3,5,7,8,9,11)$$ $$s_2 = (1 \times 10^3, 1\times 10^2, 1\times 10^1, 0.5\times 10^1, 1\times 10^0, 1\times 10^{-1}, 1\times 10^{-2})$$ that are fairly obviously arithmetic and exponential, respectively. What's the best simple rule for classifying $s_1$ as "most likely a linear sequence" and $s_2$ as "most likely a geometric sequence"?
Is there something more efficient than regressing $s_1$ and $\log(s_1)$ on $1,2,3,4,5,6,7$ and comparing $R^2$ values (and similarly for $s_2$)? Is there a standard way to classify a sequence as "more geometric than linear" or vice versa? As in the case of $$ s_3 = (2,3,4,0.5 \times 10^1, 1 \times 10^1, 1 \times 10^2, 1 \times 10^3) $$ How might we use arithmetic means and geometric means and variances?