Question

Prove that n²+2 cannot be an A.P. for every natural number.​

Answer 1

Answer:

In an A.P there is a common difference.

Proof by contradiction: Claim the statement is true and prove it's false so that the assumption is wrong.

Solution.

Let be the sequence .

Then, and .

Let's assume to the contrary that is an A.P. Then, .

Which means by , and by .

If there is a common difference, it would be . But the equation is false for all . Therefore cannot be an A.P for any .