14.1.1. Lecture -14

Try these first

1. 1.

   State the arithmetic mean of the numbers 4 and 16.

   Hint

   Add the numbers and divide by 2.

2. 2.

   Write the binary representation of 12.

   Hint

   What powers of 2 do sum to 12?

3. 3.

   What is the arithmetic mean of 6 and 14?

   • 7
   • 10
   • 11

   Hint

   Calculate the average.

4. 4.

   True or False: Every positive integer can be expressed as a sum of distinct powers of two.

   • True
   • False

   Hint

   Think about how binary numbers work.

5. 5.

   Prove by induction that for n = 2^k, the arithmetic mean is always greater than or equal to the geometric mean.

   Hint

   Refocus on properly utilizing A and G through each inductive verification.

6. 6.

   Given that 8 guests know each other, can you prove that there is no celebrity?

   Hint

   Break down the definitions clearly to find contradictions.