What's the base case when we prove inductively that 3n + 1 is divisible by 2 for all positive integers?
Hint
Choose the simplest positive integer you can think of, then plug it in for n.
Answer
When n = 1, we get 31 + 1 = 4, which is divisible by 2.
Example 2
What's the base case when we prove inductively that 10n – 5 is divisible by 5 for all positive integers?
Hint
Use the smallest possible n-value. No need to get fancy.
Answer
When n = 1, we get 101 – 5 = 5, which is divisible by 5.
Example 3
What's the base case when we prove inductively that n – 2 is positive when n > 2?
Hint
We can't use n = 1 this time, so what's the next simplest value we can plug in? Whole numbers are easier than fractions/decimals.
Answer
When n = 3, we get 3 – 2 = 1, which is positive.
Example 4
Brandon is trying to prove that 3n + 1 is an even number whenever n is a positive integer. You want to mess up his proof, because you think he's wrong. Name one counterexample that shows he can't prove his general statement.
Hint
A counterexample is just any n-value that makes the statement untrue.
Answer
When n = 2, we get 3(2) + 1 = 6 + 1 = 7, which is not an even number. (There are lots of other possible counterexamples, too.)
Example 5
Are there any counterexamples that disprove the statement, "6n + 2 is always divisible by 4 when n is a positive integer"?
Hint
It works fine when n = 1, but pay close attention to other possible values for n.
Answer
When n = 2, we get 6(2) + 2 = 12 + 2 = 14, which is not divisible by 4. (There are lots of other possible counterexamples, too.)