The fundamental theorem of arithmetic (FTA) can be proved using the following:
if a statement is true for $n=1$, and its truth for $n=1,2,\ldots,k$ implies its truth for $n=k+1$, then it is true for all $n\ge1$,
which is variously known as "strong" or "extended" induction, or the "second principle" of induction. While any such proof can be rewritten so as to use only "basic" induction (truth for $k$ implies truth for $k+1$), in the case of the FTA it is not very convenient to do so.
Question: can anyone suggest further examples of this type? I know the question is a bit vague, but trying to clarify: my main requirement is that the "strong" induction proof should not be conveniently reducible to a "basic" induction proof.
Conditions:
- Content should be accessible to first-year tertiary students.
- I have the following examples already: fundamental theorem of arithmetic; decomposition of $n$-gon into $n-2$ triangles; expression of rational numbers as sums of distinct unit fractions; expression of rational numbers as continued fractions; this nice example; equalities or inequalities for a recurrence of the form $u_{n+1}=d_1u_n+\cdots+d_nu_1$ where the $d_k$ are more or less random numbers, e.g., digits of $\pi$.
Note that in the last example, if the $d_k$ are not random, then the formula can frequently be reduced to a simpler recurrence and the problem is therefore (for present purposes) "trivial". For example, $$u_{n+1}=u_n+2u_{n-1}+3u_{n-2}+\cdots+nu_1$$ reduces to $u_{n+1}=3u_n-u_{n-1}$ which can be approached by means of a "two-step" induction.
