12.2.4. Base Case and Inductive Step

Try these first

1. 1.

   What is the base case in proof by induction?

   Hint

   Think about where we start in proving our statement.

2. 2.

   State the two main components of proof by induction.

   Hint

   Both terms start with the letter 'b' and 'i'.

3. 3.

   What are the two main components of proof by induction?

   • Inductive case and base case
   • Base case and inductive step
   • Base case and conclusion

   Hint

   Both terms depict the foundational aspects of a proof.

4. 4.

   True or False: Strong induction assumes only the property for k to prove k+1.

   • True
   • False

   Hint

   Reflect on what 'strong' means in this context.

5. 5.

   Prove by induction that for all n >= 1, the sum of the first n even numbers is n(n + 1).

   Hint

   Use the formula for the sum of the first k even numbers.

6. 6.

   Use strong induction to prove that every positive integer greater than 1 is either prime or can be factored into primes.

   Hint

   Examine the factors of k + 1 in relation to those below it.