Mathematical induction is used in mathematics to prove many
statements, in particular it is used to prove statements, theorems, or even formulas
that are asserted by all natural numbers. When we say “all” natural numbers it
means any natural number that we may possibly come across on. In order to prove
by mathematical induction, we must first go over some very important rules.
To prove a statement by induction, the first step that must
be done is to prove part one. According to part one, if this is true then step two is also true. After applying step two we are now implying that the statement
will be true for \(n=2,3,4\), and so on.
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.