Exercise 1.13: Prove that 
  
    Fib
    (
    n
    )
  
 is
the closest integer to 
  
    
      φ
      n
    
    
      /
    
    
      5
    
  
, where 
  φ
  =
  
    (
    1
    +
    
      5
    
    )
    
      /
    
    2
  
.  
Hint: Let 
  ψ
  =
  
    (
    1
    −
    
      5
    
    )
    
      /
    
    2
  
.
Use induction and the definition of the Fibonacci numbers (see 1.2.2) 
to prove that 
  
    Fib
    (
    n
    )
  
  =
  
    (
    
      φ
      n
    
    −
    
      ψ
      n
    
    )
    
      /
    
    
      5
    
  
.