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 __________________________________________________________
   
   
   ______________________________________________________________________________.
   

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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!)