Show that n2 n is divisible by 2 for all natural numbers NS

Show that n^2 + n is divisible by 2 for all natural numbers N

Solution

We have n²+ n = n ( n + 1). The set N of whole numbers { 0,1,2,3,4,5,…….} is called the set of natural numbers. We shall consider three cases:

Case 1:

When n = o, we also have n² + n = 0. We know that o is divisible by 2. Therefore, n² + n is divisible by 2.

Case 2:

When n is an even number. Then nis divisible by 2, so that n² + n = n (n + 1) is divisible by 2.

Case3:

When n is an odd number. Then n + 1 is an even number and is, therefore, divisible by 2. Then n² + n = n(n + 1) is divisible by 2.