Show that is an irrational number, given that 7 is irrational.

OR

Prove that n² + n is divisible by 2 for any positive integer n.

Let us assume is rational

can be written in the form where a and b are co-prime.

Hence,

Since, rational ≠ irrational

This is a contradiction.

, Our assumption is incorrect.

Hence, is irrational.

OR

Case I: Let n be an even positive integer.

Then, n = 2q, we have

n² + n = (2q)² + 2q

= 4q² + 2q

= 2q (2q + 1)

⇒ n² + n = 2r, where r = q (2q + 1)

⇒ n² + n is divisible by 2 .

Case II: Let n be an odd positive integer.

Then, n = 2q + 1

⇒ n² + n = (2q + 1)² + (2q + 1)

= 4q² + 1 + 4q + 2q + 1

= 4q² + 6q + 2

= 2(2q² + 3q + 1)

⇒ n² + n = 2r, where r = 2q² + 3q + 1

⇒ n² + n is divisible by 2.

∴ n ² + n is divisible by 2 for every integer n.

Hence proved