Question

prove that n2+n is divisible by 2 for any positive integer n
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Answer 1

Step-by-step explanation:

To prove:- n²+n is divisible by 2 for any positive integer n.

Method 1:-

Case 1:-If n is even

Let n=2k

n²+n

n(n+1)

2k(2k+1)

It is divisible by 2 bcz it's a multiple of 2.

Case 2:-If n is odd

n=2k+1

n²+n

n(n+1)

(2k+1)(2k+1+1)

(2k+1)(2k+2)

(2k+1)(2)(k+1)

This expression is a multiple of 2

So,it's divisible by 2.

For both the cases,n is divisible by 2.So for any positive integer n,n²+n is divisible by 2.

Hence proved

Method 2:-

n²+n

n(n+1)

n and n+1 are 2 consecutive positive integers

We know that the product of p consecutive integers is divisible by p!.

So the product of 2 consecutive integers is divisible by 2!.

Product of the 2 consecutive integers is divisible by 2.

Here n and n+1 are 2 consecutive positive integers.

So n(n+1) is divisible by 2.

Hence proved

Note:

n!=n(n-1)×(n-2)×(n-3)....... ×3×2×1