Thread: Increasing sequence

Is it possible to show that the sequence defined by
$\displaystyle a_1 = 1, \hskip 2em a_{n+1} = 3-\frac{1}{a_n}$
is increasing without relying on mathematical induction?

Originally Posted by Euclid's Bridge Habitant

Is it possible to show that the sequence defined by
$\displaystyle a_1 = 1, \hskip 2em a_{n+1} = 3-\frac{1}{a_n}$
is increasing without relying on mathematical induction?

Almost certainly not, because the result depends on knowing that $\displaystyle a_1=1$. If for example $\displaystyle a_1=4$ then the sequence will be decreasing. In order to use information about $\displaystyle a_1$ to prove something about $\displaystyle a_{n+1}-a_n$, 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 $\displaystyle a_1=1$. If for example $\displaystyle a_1=4$ then the sequence will be decreasing. In order to use information about $\displaystyle a_1$ to prove something about $\displaystyle a_{n+1}-a_n$, 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.