Question

Prove that (n2 + n) is divisible by 2 for any positive integer n.
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Answer 1

Step-by-step explanation:

Case i: Let n be an even positive integer.

When n=2q

In this case , 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.

When n=2q+1

In this case 

n2−n=(2q+1)2−(2q+1)=(2q+1)(2q+1−1)=2q(2q+1)

n2−n=2r, where r=q(2q+1)

n2−nis divisible by 2.

∴  n2−n is divisible by 2 for every integer n

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Answer 2

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

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