TITLE: What is the second principle of finite induction?
QUESTION [22 upvotes]: I understand the principle of finite induction, but my book then mentions that there is a variant of the first where requirement b is changed to 

If $k$ is a positive integer such that $1,2, \dots, k$ belong to $S$, then $k + 1$ must also be in $S$.

The sample problem is proving that the inequality about the Lucas numbers 
$l_n < (7/4)^n$.
I understand the algebra of the proof, but I do no understand how it uses the second principle or how it is different from the first principle?
The proof for the lucas numbers
$$
a_1 = 1 < \left(\frac{7}{4}\right)^1 = \frac{7}{4}\text{ and }a_2 = 3 \lt \left(\frac{7}{4}\right)^2 = \frac{49}{16}.$$
Choose an integer $k\geq 3$ and assume that the inequality is valid for $n = 1,2,\ldots,k$.  Then
$$a_{k-1} \lt \left(\frac{7}{4}\right)^{k-1}\text{ and }a_{k-2} \lt \left(\frac{7}{4}\right)^{k-2},$$
so 
\begin{align*}
a_k &= a_{k-1} + a_{k-2}\\\
&\lt \left(\frac{7}{4}\right)^{k-1} + \left(\frac{7}{4}\right)^{k-2}\\\
&= \left(\frac{7}{4}\right)^{k-2}\left(\frac{7}{4} + 1\right)\\\
&= \left(\frac{7}{4}\right)^{k-2}\left(\frac{11}{4}\right)\\\
&\lt \left(\frac{7}{4}\right)^{k-2}\left(\frac{7}{4}\right)^2\\\
&= \left(\frac{7}{4}\right)^k
\end{align*}
The responses are all helpful, but if someone could point out what the difference is when you go about actually using strong induction instead of weak induction. (You could use the lucas number proof or something else)

REPLY [7 votes]: There is a common misconception that strong induction is somehow "stronger" than normal induction, but it isn't really. Strong induction is just the special case of normal induction in which the induction hypothesis $P(n)$ takes the form "for all $m < n$, $Q(m)$". Thus strong induction is actually a limited form of the principle of induction, because not all induction hypotheses are of that special form. On the other hand, any proof by strong induction can be trivially rephrased as a proof by "weak" induction. 
One reason for the terminological difficulty is that the only place that people talk about "strong induction" is in introductory courses. There, "use strong induction" can be a hint about what sort of induction hypothesis to choose. In these classes, the principle of induction is stated somewhat informally, so it may not be obvious what counts as a legal induction hypothesis in the first place. 
In contexts where the principle of induction is stated formally, e.g. Peano arithmetic, nobody talks about strong induction at all. The axiom scheme for induction in Peano arithmetic simply includes all axioms of the form
$$
(P(0) \land (\forall n)(P(n) \to P(n+1)) \to (\forall n)(P(n))
$$
where $P$ is any formula of Peano arithmetic. This includes formulas of the form $(m < n)\to Q(m)$ as well as all other formulas.