14.2.2. Proceeding with Induction

Try these first

1. 1.

   What is the arithmetic mean of 3 and 9?

   Hint

   Add the two numbers together and divide by 2.

2. 2.

   Give an example of a positive integer and its binary representation as a sum of distinct powers of two.

   Hint

   Remember each integer has a unique binary form.

3. 3.

   What is the formula that defines the relationship between AM and GM?

   • AM = GM
   • AM ≥ GM
   • AM ≤ GM

   Hint

   Think of how averages behave.

4. 4.

   True or False: Every positive integer has a unique binary representation as a sum of distinct powers of two.

   • True
   • False

   Hint

   Consider how binary numbers correlate to decimal.

5. 5.

   Using induction, determine a formula for proving that the sum of the first n odd numbers equals n^2.

   Hint

   Sum the k odd numbers and add the next to the result.

6. 6.

   Give a rigorous proof that the infinite series 1/2^n converges, using bounds established via induction.

   Hint

   Use geometric series properties.