What is mathematical induction?
Mathematical induction is a powerful method used to establish that a given statement
(i.e. a theorem or a formula) is TRUE for all natural numbers.
Note: The principle of mathematical induction is in fact a fully rigorous and deductive
form of reasoning, not to be confused with inductive reasoning.
The method of a proof by mathematical induction goes as follows:
Let S (n ) be a statement which holds for any natural n (i.e. n  1, 2,3,... )
If
1)
S (1) is TRUE, and
2)
S (k  1) is TRUE whenever S ( k ) is TRUE,
then
the statement S (n ) is TRUE for all n  {1,2,3,...} .
The first step is called the base case and the second step is called the inductive step.
The assumption in the inductive step that S ( k ) is true is called the induction
hypothesis.
Example
Below, we prove that the sum of the first n natural numbers is given by the formula
S ( n)  1  2  3    n 
1)
n  (n  1)
.
2
Base Case
S (1)  1 
1 2
.
2
The formula is (trivially) TRUE for n  1.
2)
Inductive Step
First we assume that S ( k ) is TRUE. We then have
S (k )  1  2  3 
k 
k  (k  1)
.
2
This is the induction hypothesis.
Now we use this hypothesis to prove that S (k  1) is TRUE, or that
S (k  1) 
(k  1)  (k  2)
.
2
Adding an additional term to the sum of the first k natural numbers, namely
k  1 , yields:
S (k  1)  1  2  3 
 k  (k  1)
 S (k )  (k  1)

k  (k  1)
 (k  1)
2

k  (k  1)  2(k  1)
2

(k  1)  ( k  2)
2
This finalizes the proof by induction.
