Thread: Increasing sequence

Is it possible to show that the sequence defined by

is increasing without relying on mathematical induction?

Originally Posted by Euclid's Bridge Habitant

Is it possible to show that the sequence defined by

is increasing without relying on mathematical induction?

Almost certainly not, because the result depends on knowing that . If for example then the sequence will be decreasing. In order to use information about to prove something about , you need to bridge the gap between 1 and n. The only apparent way to do this is to use an inductive method.

Originally Posted by Opalg

Almost certainly not, because the result depends on knowing that . If for example then the sequence will be decreasing. In order to use information about to prove something about , you need to bridge the gap between 1 and n. The only apparent way to do this is to use an inductive method.

I might add that if you could solve the difference equation eactly, you might be able to deduce that it is increasing but this is a nonlinear difference equation and they tend to be very difficult to solve exactly.