12.2.5. Common Mistakes in Proof by Induction

Try these first

1. 1.

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

   Hint

   Think about where the induction starts.

2. 2.

   What is the inductive step in induction?

   Hint

   What transition do we make from k to k+1?

3. 3.

   What is the first step in a proof by induction?

   • Prove the base case
   • Prove the inductive step
   • Assume P(k) is true

   Hint

   What step do we always begin with in the induction framework?

4. 4.

   True or False: Strong induction requires proving the property for all preceding integers.

   • True
   • False

   Hint

   What makes it 'strong' compared to regular induction?

5. 5.

   Prove by induction that for all integers n, 3^n > n^3 for n >= 3.

   Hint

   Use properties of exponential functions and inequalities.

6. 6.

   Show using strong induction that every integer greater than 1 can be expressed as a product of prime numbers.

   Hint

   Consider prime factorization and how primes build numbers.