Mathematical Induction – Miller College Algebra Section 8.4
Suppose that we are trying to prove that a statement is true for all positive integers.
Step 1:
Show that the statement is true when n = 1 (existence step).
Step 2:
Show that if the statement is true for n = k, then it must be true for n = k + 1 (inductive step)
This implies that if the statement is true for n = 1, then it must be true for n = 2; if it is true for n
= 2, then it must be true for n = 3, etc.
Step 3:
Write the conclusion – the statement is true for all positive integers.
Example –
Show that 3  4  5 
  n  2 
n  n  5
2
Step 1:
n = 1: 3 
11  5
2
.
Step 2:
Assume that 3  4  5 
Show that 3  4  5 
  k  2 
k  k  5
2
  k  2     k  1  2  
 k  1   k  1  5
2
k  k  5
 k  1 k  6 
k 3
2
2
k 2 5k
k 2  7k  6
7k
 k 3
 k2 
 3.
2
2
2
2
Step 3: Therefore 3  4  5 
  n  2 
n  n  5
is true for all positive integers.
2