Question

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

Answer 1

Answer:

Case i: Let n be an even positive integer.

When n=2q

In this case , we have

n

2

−n=(2q)

2

−2q=4q

2

−2q=2q(2q−1)

n

2

−n=2r , where r=q(2q−1)

n

2

−n is divisible by 2 .

Case ii: Let n be an odd positive integer.

When n=2q+1

In this case

n

2

−n=(2q+1)

2

−(2q+1)=(2q+1)(2q+1−1)=2q(2q+1)

n

2

−n=2r, where r=q(2q+1)

n

2

−nis divisible by 2.

∴ n

2

−n is divisible by 2 for every integer n