Say whether the following is true or false. If true, prove using mathematical induction. If false, give a counter-example. 3^2n-4 is a multiple of 5
Actually the bot screwed up with its counterexample, even though it
reached the right conclusion
The AI said:
When n=2, 3^(2(2))-4 = 65 which is not a multiple of 5
it should have been:
When n=2, 3^(2(2))-4 = 76 which is not a multiple of 5
To determine whether 3^(2n - 4) is a multiple of 5, we will use mathematical induction.
Step 1: Base case
Let n = 1.
For n = 1, we have 3^(2*1 - 4) = 3^(-2) = 1/9, which is not a multiple of 5. Therefore, the statement is false for n = 1 as a counter-example has been found.
Alternatively, we can check the potential counter-example value for n = 1:
3^(2*1 - 4) = 3^(-2) = 1 / (3^2) = 1 / 9
Since 1/9 is not divisible by 5, the statement is false for n = 1.
Therefore, we do not need to proceed to the inductive step as the statement has been proven false.
To determine whether the statement "3^(2n-4) is a multiple of 5" is true or false, we can use mathematical induction.
First, let's check the base case when n = 1:
When n = 1, the expression becomes 3^(2(1)-4) = 3^(-2) = 1/3^2 = 1/9. Since 1/9 is not a multiple of 5, the base case is false.
Therefore, we have found a counter-example, and the statement is false.
The statement is false.
Counter-example:
When n=1, 3^(2(1))-4 = 5 which is a multiple of 5
When n=2, 3^(2(2))-4 = 65 which is not a multiple of 5
Therefore, the statement is false since it is not true for all values of n.
Counter-example is the simplest way to prove that a statement is false.