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CMSC 250

Quiz #9

Wed., Mar. 29, 2006

Write all answers legibly in the space provided. The number of points possible for each question is indicated in square brackets - the total number of points on the quiz is 30, and you will have exactly 20 minutes to complete this quiz. You may not use calculators, textbooks or any other aids during this quiz.

1. [15 pnts.] Prove the following statement is true or give a specific counter example to prove that it is false.

   
   

   $$\forall n \in Z^{\geq 4}, \,\,\, 2^n \geq n^2$$
   
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   $\downarrow$ TURN OVER $\downarrow$

2. [15 pnts.] Prove the following statement is true or give a specific counter example to prove that it is false:
   
   $$\forall n \in Z^+, \,\,\, \sum_{j=0}^{n} [3(2^j)] = 6(2^{n}) - 3$$
   
   

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