SOLUTION: Using direct proof, prove that if n is a natural number, then n(n+1) is even.

Question 1045633: Using direct proof, prove that if n is a natural number, then n(n+1) is even.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i'm not real good at these, but let me give it a shot.

you want to prove that n * (n + 1) is even, when you are dealing with a natural number.

a natural number is all the positive integers i believe.

it does not include 0 as far as i remember.

you have 2 possibilities.

if n is even, then n + 1 is odd.

if no is odd, then n + 1 is even.

assuming that n is even, then there exists a number k such that 2 * k = n.

n * (n + 1) becomes 2 * k * (2 * k + 1) which becomes 4 * k^2 + 2 * k.

the number 4 * k^2 + 2 * k has to be even because it's divisible by 2.

assuming that n is odd, then there exists a number k such that 2 * k = n + 1.

n * (n + 1) becomes n * 2 * k which becomes 2 * k * n.

the number 2 * k * n has to be even because it's divisible by 2.

any number that's divisible by 2 is even.

not sure if this constitutes an acceptable proof by your instructor, but it seems reasonable to me.

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