Fill-in-the-blank Induction Exercises

  1. Use mathematical induction to show that 3 + 4 + ... + n-2 = (n+1)(n-4)/2, n>=5.
    Base case: n=____.  Left side:________________  Right side:__________________
    
    Inductive step: 
    
    If _____________________________________  then ____________________________________.
             (property for k)                         (property for k+1)
    
    
                                               => ____________________________________
                                                      (simplify property for k+1)
    
    Proof:
    By the inductive hypothesis, ________________________________________________.
    
    To make the left side of this look like what we are trying to show, add
    
     ___________ to both sides.
    
    
    This gives ___________________________________________________________________.
    
    
    Simplifying, we get __________________________________________________________
    
    
    ______________________________________________________________________________.
    


  2. Use mathematical induction to show that 1 + 2 + ... + n+1 = (n+1)(n+2)/2
    Base case:
    
    
    Inductive step:
    
    
    
    
    Proof:
    By the inductive hypothesis, _________________________________________________.
    
    (... You finish!)