> learn to think in terms of induction hypotheses.

Can you expand on this or link to a good definition?

The induction hypothesis is a process in mathematics which assumes a formula for P(n) which is true, then uses this assumption and the problem definition to prove the case P(n+1).

As a concrete example, take the formula for the sum of integers: S(n)= n * (n+1)/2. Proving this inductively means asking what value should be assigned to S(n+1). We can calculate this either with a direct "change of variable" expansion of the hypothesized formula or, nothing that S() is the sum of a series, there is an alternate expansion of S(n)+(n+1). Refactoring both sides yields the same algebraic result, (n+1) * (n+2)/2.

[The final step for a full proof would be to establish a "baseline" case, traditionally either 0 or 1, where we show our formula yields the correct result. The inductive step, applied repeatedly, then proves the formula must be correct for the next higher input, and the one just after that, ad inifinitum.]