Constant differences and ratios

An arithmetic progression adds the same difference d at each step. A geometric progression multiplies by the same ratio r. Compare several successive differences or ratios, not just the first pair. 7,11,15,19, has difference 4. The nth term is reached after n-1 additions, so u_n=7+4(n-1). 81,27,9,3, has ratio 13. After n-1 multiplications, u_n=81(13)n-1. For first term a, an AP has u_n=a+(n-1)d; a GP has u_n=arn-1. A negative difference means repeated subtraction; a negative ratio alternates signs. 2,4,8,14 is neither: differences 2,4,6 vary and ratios 2,2, 74 vary. A constant nonzero sequence is both an AP with d=0 and a GP with r=1. In a context, decide whether each step adds a fixed amount or changes by a fixed percentage. Check which event is term one before inserting an index.