Question

Identify the correct proof by induction or a counterexample to disprove the following statement.

Statement: For all positive integers n, 3^n < 2^n.

a) Proof by induction
b) Counterexample
c) Both a and b
d) None of the above

Answer 1

The statement 'For all positive integers n, 3^n < 2^n' is disproven by the counterexample n=1, where 3^1 is not less than 2^1. Thus, a counterexample is the correct method to disprove the given statement.

Step-by-step explanation:

The statement that for all positive integers n, 3^n < 2^n can be easily disproven by providing a counterexample. If we look at the smallest positive integer, which is 1, we see that 31 is not less than 21. Both of them are equal to 3 and 2, respectively, so the given statement is false.

Counterexamples are useful for disproving statements like these. By demonstrating a single case where the statement does not hold, the assertion that the statement is true for all positive integers n is invalidated. Thus, the correct answer to disprove the statement given is (b) Counterexample.