1549901369-Elements_of_Real_Analysis__Denlinger_

26 Chapter 1 • The Real Number System

*Proof. Exercise 18. •

Similarly, the Alternate Principle of Mathematical Induction can be rewrit-
ten to start with any natural number.

MATHEMATICAL INDUCTION

AS A METHOD OF DEFINITION

When we wish to define a quantity f(n), for all natural numbers, math-
ematical induction is often useful. This method of definition is often called
"recursive" definition, especially in computer science. Consider the following
example:

Definition 1.3.12 We define an, V n EN, as follows:

(1) a^1 =a;

(2) Vk E N, ak+^1 = a · ak.

EXERCISE SET 1.3

l. Prove Theorem 1.3.8 (c).

1 1. 1 1

2. Prove that Vn EN, 0 < 2 :::; - :::; 1; 1f n > 1, then 0 < 2 < - < l.
   n n n n
   In Exercises 3-22, use mathematical induction to prove the given equa-
   tion, statement, or inequality, Vn EN.


3. l+2+3+···+n=n(n+l).
   2

12 22 32 2

n(n + 1)(2n + 1)

4. 
   
   
   
   • 
     
     
     
     • 
       
       
       
       • ··· +n =
         6
         n^2 (n + 1)^2
       
       
       
       
     
     
     
     
   
   
   
   


5. 
   

   13 + 2^3 + 3^3 + · · · + n^3 = ----
   4

   
   


6. 
   

   1+3+5+· · ·+(2n-l)=n^2.

   
   


7. 
   

   1+4 + 7 + · · · + (3n - 2) = n(
   3
   ~ - l)

   
   


8. 
   

   n(n + l)(n + 2) is divisible by 3.

   
   


9. 
   

   n^5 - n is divisible by 5.

   
   


10. 
    

    1 + ~ + i + ~ + ... + 2~ = 2 - 2~.

    
    


11. 
    

    1 + l + 3 l + 9 ...1. 27 + ... + 3n ...l.. = ~ 2 - ~ 2 3 (~)n