14.2. Proof by Induction

Try these first

1. 1.

   What are the two main steps in a proof by induction?

   Hint

   Think about the first step in proving something is true.

2. 2.

   What is the base case for proving that the arithmetic mean is at least the geometric mean for 2 numbers?

   Hint

   Consider two positive integers and their means.

3. 3.

   What is the purpose of the base case in proof by induction?

   • To prove the statement for all integers
   • To establish the truth for one initial case
   • To derive a general formula

   Hint

   Remember that every proof needs a starting point to build upon.

4. 4.

   A celebrity in a party knows everyone but is not known by anyone.

   • True
   • False

   Hint

   Think about the social dynamics of knowing someone.

5. 5.

   Prove using induction that for every n > 1, n! > 2^n.

   Hint

   Consider how quickly factorial values grow compared to powers!

6. 6.

   Prove that for any integer n, n² + n is even.

   Hint

   Factor to see the evenness clearly.