This is related to the Previous note about Induction
Out of the problems stated in the note, to and are standard results, you can use them like identities.
Here's the proof of which is slightly different than other ones.
Problem is to prove that ......
Observe that for , we have . Hence the result is true for .
Assume that the result is for , and we'll prove for
For this, we use a .
Consider the term .
This term is equivalent to ...... this is because and this term we want in our proof!!!
Returning to the proof, we have assumed that the result is true for , hence in the term, i.e.
Thus what only remains to prove is is divisible by 24, see that
. So the problem has reduced to proving or even simpler,
We know that
Hence . Hence the other term in the expression is also divisible by 24, thus if the result is to be true for then we are it to be true for and for we already proved, thus it's true for natural numbers.