Question

Say whether the following is true or false. If true, prove using mathematical induction. If false, give a counter-example. 7^2n-9 is a multiple of 2

Answer 1

Looks like the bot assumed you meant:
7^(2n) - 9 , and then its conclusion is wrong.
(it actually screwed up in its counterexample stating that 40 is not a multiple of 2, lol )

Easy to prove:

7^2n yields multiples of 7
all multiples of 7 are odd
any odd number - 9 is even, so

7^(2n) - 9 is always even, and all even numbers are multiples of 2

Answer 2

The statement is false.

Counter-example: When n = 1, we get

7^2(1) - 9 = 49 - 9 = 40

which is not a multiple of 2.

Therefore, the statement is false.