I have a question regarding strong and weak induction.
During my studies only weak induction has been formally introduced but at some point I started using strong induction whenever I had something to prove for positive integers. I have some concerns about how to show to my students the difference between two of them and when is one better than another. See this example
(Exercise 19 from this book) Show that each integer $n>7$ can be written as a sum containing only the numbers $3$'s and $5$'s. For example: $8=3+5$, $9=3+3+3$, $10=5+5$.
My proof (strong induction): Base cases are already shown in the statement. Suppose that the statement is true for all numbers $11\leq n\leq k$. Now, let us observe number $n=k+1$. We have $$k+1=(k-2)+3,$$ and $k-2$ can be written with only $3$'s and $5$'s by the inductive assumption (and the base case) since $k-2\geq 9$. So we have proven our statement.
My concern is if my use of strong induction is wrong in this case in some way, or just not "pretty enough" for this problem?
Is there any reason why it is not good idea to tell students of introductory class in number theory that they can use strong induction whenever they feel they need it? Is it wrong to write assumption as in the strong case every time, even if you use only $k$ to prove $k+1$?