Quick questions on Mathematical induction for general statements: recurrence relations and properties

A closed form for a recurrence is a direct formula $a_n = g(n)$ that gives the $n$-th term without computing all the earlier ones. You are typically given a starting value $a_1 = c$ and a rule $a_{n + 1} = f(a_n)$, and asked to prove the formula $a_n = g(n)$ holds for all $n$. The strategy:

A property preserved by a process is one that, once true at some stage, stays true at the next. Many questions describe an iteration (repeatedly applying an operation, building up a figure, running an algorithm) and ask you to prove some feature holds at every stage. The inductive step is a conditional claim: if the property holds after $k$ steps, it holds after $k + 1$. You assume the property at stage $k$, perform one more step of the process, and show the property survives.

Strong induction is the variant where the step at $n = k + 1$ may rely on the statement at all earlier values, not just at $n = k$. It is occasionally useful when a recurrence reaches back two terms, such as the Fibonacci rule $F_{n + 1} = F_n + F_{n - 1}$, since the step then needs both $F_k$ and $F_{k - 1}$. Standard induction is enough for almost all HSC Extension 1 questions. If you do need the stronger form, say so: state the hypothesis as "assume the statement holds for all integers up to $k$" and verify enough base cases (two of them for a two-term recurrence) to get the chain started.