Which of these formulas does NOT define an arithmetic progression?

A sequence is an AP exactly when $a_n$ is a linear expression in $n$, because then $a_n-a_{n-1}$ is constant. For $a_n=n^2+1$ the terms are 2, 5, 10, 17, ... with gaps 3, 5, 7 - not constant, so it is not an AP. The other three are linear, with common differences -1, 4/5 and 2 respectively.