Weak and Strong Induction (examples of the "strong" sort?)

[apple2357]

OK, I have googled this but can't seem to find a simple explanation of the difference.
My understanding of weak induction is the induction that is taught in schools - i.e. Check P(1) is true, Assume P(k) is true, prove P(k+1) etc.
Can anyone offer an accessible example of strong induction?

 

[Dr.Peterson]

OK, I have googled this but can't seem to find a simple explanation of the difference.
My understanding of weak induction is the induction that is taught in schools - i.e. Check P(1) is true, Assume P(k) is true, prove P(k+1) etc.
Can anyone offer an accessible example of strong induction?

Did you try googling "strong induction example"? You should be able to find plenty. Here are a couple pages I found:

https://brilliant.org/wiki/strong-induction/
https://en.wikipedia.org/wiki/Mathematical_induction#Complete_(strong)_induction

Basically, strong induction requires a stronger hypothesis; it is used when that makes a proof easier. Since the two are equivalent, it is never necessary to use one or the other, but often one is more natural than the other.

 

[apple2357]

Thanks very much. I was searching for an explanation of the difference!
But thank you

 

[Dr.Peterson]

Thanks very much. I was searching for an explanation of the difference!
But thank you

I thought I did explain the difference, in addition to providing examples as you asked. (And the articles said more about it.)

Is there something you still lack?

 

[apple2357]

I thought I did explain the difference, in addition to providing examples as you asked. (And the articles said more about it.)

Is there something you still lack?

No you did! Thanks. When i originally googled i couldn't find an explanation. But you have helped thanks

 

[HallsofIvy]

For others who are wondering and don't want to go to the enormous trouble of looking it up:
"Weak induction" says that if statement P(n), depending on the positive integer, n, has the properties that
1) P(1) is true
2) Whenever P(k) is true, P(k+1) is true
then P(n) is true for all positive integers, n.

"Strong induction" says that if statement P(n), depending on the positive integer, n, has the properties that
1) P(1) is tru
2) Whenever P(i) is true for all \(\displaystyle i\le k\) P(k+1) is true
then P(n) is true for all positive integers, n.