Download Sullivan College Algebra Section 9.4

Sullivan Algebra and
Trigonometry: Section 13.4
Objectives of this Section
• Prove Statements Using Mathematical Induction
Theorem The Principle of Mathematical
Induction
Suppose the following two conditions are satisfied
with regard to a statement about natural numbers:
CONDITION I: The statement is true for the natural
number 1.
CONDITION II: If the statement is true for some
natural number k, it is also true for the next natural
number k + 1.
Then the statement is true for all natural numbers.
n(n  1)
Show that 1  2  3   n 
2
is true for all natural numbers n.
CONDITION I: Show true for n = 1
1(1  1) 1(2)

1
2
2
CONDITION II: Assume true for some
number k, determine whether true for k + 1.
k ( k  1)
Assume: 1  2  3  k 
2
( k  1)( k  2)
Show: 1  2  3  k  ( k  1) 
2
( k  1)( k  2)
1  2  3  k  ( k  1) 
2
k ( k 1)
2
k ( k  1)
k ( k  1) 2( k  1)
 ( k  1) 

2
2
2
2
k
 3k  2
k  k  2k  2


2
2
2
( k  1)( k  2)

2