Q:

Strong explanation of Strong Form of Mathematical Induction

I don't quite understand induction well, and was wondering if you could explain to me what induction is and what the strong form of induction is. 

A:

In regular induction, you assume some property holds true at some arbitrary index value, say $k$. You might not know whether it does or doesn't, but you say, "let's imagine it does, and see what happens."
Then, given your assumption at that value, you need to show that it must hold true at the next value, $k+1$. In other words, a consequence of your assumption is that the property holds true for the next value in your set.
But, you must always prove the base case as well, so you have to have a sort of anchor to say "it definitely holds at this base value, therefore, I have a foundation to build on." Often times, the base case is $k=1$, but not always. Sometimes it makes more sense to start at $k=2$, or $k=50$, or whatever. It all depends on your proof.
Strong induction is the same thing, except instead of assuming the property holds at some single arbitrary index value $k$, you assume that it holds for a range of them $k, k+1, k+2, \ldots, k+n$.

