14.2.1. Base Case and Inductive Hypothesis

Try these first

1. 1.

   What is the base case in proof by induction?

   Hint

   Think about the smallest natural number applicable.

2. 2.

   What does the inductive hypothesis assume?

   Hint

   Consider what you need to prove next.

3. 3.

   What is the first step in proof by induction?

   • Establishing the base case
   • Formulating the hypothesis
   • Proving for k + 1

   Hint

   Think about where you start before building upon your argument.

4. 4.

   True or False: The inductive hypothesis is the part of the proof where we assume the statement is true for k.

   • True
   • False

   Hint

   Consider what is required to build upon your assumption.

5. 5.

   Prove that the formula for the sum of the first n odd numbers is n^2 using induction.

   Hint

   Focus on the algebraic manipulation of odd numbers.

6. 6.

   Show that for any integer n >= 1, 2^n - 1 is divisible by n.

   Hint

   Factorize and explore the properties of exponential numbers.