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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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