In its most simple and general form the proof by recurrence of a formula containing constants and one variable consists in giving a number of successive integral values to the variable, and showing by trial that the formula is true for those values. The next step is to show that the form of the result remains the same when n+1 is substituted for n, or that each result depends upon the preceding one. If this second step is neglected, as it often used to be, the proof seems to be invalid.

This proof by mathematical or successive induction has been very generally accepted by mathematicians and neglected by logicians.