One of the standard proofs for the Arithmetic Mean-Geometric Mean Inequality is via the following induction route:
Let $n$ be the number of (non-negative) variables. It is proved for $n=1,2$ (straightforward), and then it is proved that if it is valid for $n \ge 2$ then it is valid for $2n$ and $n-1$.
This implies that the inequality is valid for all $n$ which are powers of 2, and since any positive integer is less than some power of 2, the inequality follows for all $n \ge 1$.
Are the more examples of this kind of induction (backward induction or $n \to 2n$), or in general not the usual $n \to n+1$ or $1,2,\cdots,n \to n+1$ induction schemes?