Question

Suppose n is an integer. select all statements below that are true:

1) n^2 + n is always an even integer
2) n^2 + n is always an even integer when n is even
3) n^2 + n is always an even integer when n is odd
4) n^2 + n is never an even integer when n is odd
5) n^2 + n is never an even integer
6) n^2 + n is sometimes an even intege

Answer 1

Answer: 1)
`n^2 + n` is always an even integer

2)
`n^2 + n` is always an even integer when n is even

3)
`n^2 + n` is always an even integer when n is odd

Explanation:

Since, if n is an integer,

Then n = even or odd,

If n = even ⇒
`n^2` = even

⇒
`n^2+n` = even+ even = even

Now, If n = odd ⇒
`n^2` = odd ( square of a odd number is always odd )

⇒
`n^2+n` = odd + odd = even

Hence, however n is odd or even,

`n^2+n` is always an even number.

⇒ Options 1), 2) and 3) are correct.