authorea.com/101187

Rules:

1.) If when a statement, theorem, or formula is true for any natural number \(n=k\) then it is also true for \(n=k+1\)

and

2.) The statement is true for \(n=1\) then the statement will be true for every natural number \(n\).

Looking back at step one, when we say “the statement is true
for \(n=k\),” this is called an induction assumption or even an induction hypothesis.
Our hypothesis is what we assume when we are trying to prove our statement by induction.

One very important thing to remember is that when working
with induction most people will show that if \(P\left(k\right)\) is true then \(P\left(k+1\right)\) is also
true because this is the most important part about induction, it is what we
want to prove. However, a common mistake that is made is remembering to prove \(P\left(1\right)\) true, if this is not shown then we cannot know for sure that \(P\left(2\right)\), \(P\left(3\right)\), and so
on is true.

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