what is strong induction and how does it differ from regular induction.
Mathematics

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what is strong induction and how does it differ from regular induction.
just definition please
Sometimes the definition given above for induction is called ``weak''induction  as contrasted with ``strong induction'', defined as follows: If is an integer, and is some property of integers, and ``base case'': is true and. ``induction step'': whenever are all true, then is also true. then is true for all integers
With simple induction you use "if $p(k)$ is true then $p(k+1)$ is true" while in strong induction you use "if $p(i)$ is true for all $i$ less than or equal to $k$ then $p(k+1)$ is true", where $p(k)$ is some statement depending on the positive integer $k$.
They are NOT "identical" but they are equivalent.
It is easy to see that if simple induction is true then strong induction is true: if you know that statement $p(i)$ is true for all $i$ less than or equal to $k$, then you know that it is true, in particular, for $i=k$ and can use simple induction.
It is harder to prove, but still true, that if strong induction is true, then simple induction is true. That is what we mean by "equivalent".
Here we have a question. It is not why we still have "weak" induction  it's why we still have "strong" induction when this is not actually any stronger.
My opinion is that the reason this distinction remains is that it serves a pedagogical purpose. The first proofs by induction that we teach are usually things like $∀n[∑_{i=0}^n \frac{n(n+1)}{2}]$. The proofs of these naturally suggest "weak" induction, which students learn as a pattern to mimic.
Later, we teach more difficult proofs where that pattern no longer works. To give a name to the difference, we call the new pattern "strong induction" so that we can distinguish between the methods when presenting a proof in lecture. Then we can tell a student "try using strong induction", which is more helpful than just "try using induction".
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