12.2.10. Equivalence of Regular and Strong Induction

Try these first

1. 1.

   Define proof by induction in your own words.

   Hint

   Think about how it serves as a mechanism to establish truth across an infinite set.

2. 2.

   What must be shown in the base case of an induction proof?

   Hint

   Consider the first number in the sequence you're working with.

3. 3.

   What is the first step in proof by induction?

   • Establish the base case
   • Prove k + 1
   • Assume P(k) is true

   Hint

   Remember, without a strong foundation, the rest doesn't stand.

4. 4.

   In strong induction, which series of assumptions can you use?

   • P(k)
   • P(b) through P(k)
   • None

   Hint

   Think about how many cases are at your disposal!

5. 5.

   Prove that for any integer n ≥ 1, n^3 - n is divisible by 6.

   Hint

   Focus on how n^3 and n can be restructured in forms conducive to divisibility.

6. 6.

   Using strong induction, prove that any integer n ≥ 12 can be expressed as a sum of 4s and 5s.

   Hint

   Consider how the addition of either a 4 or 5 affects the total in constructing k + 1.