Had trouble falling asleep last night, so I began adding up numbers in my head. "Discovered" this interesting property of integers, though I'm sure it already exists more formally in the mathematics world:
Let n be a positive integer. Let S be the sum of all integers from 1 to n, inclusive.
If n is odd, then S is a multiple of n. However, the same property does not exist for even numbers.
It will take me some time to try to even wrap my head around this, and who knows if I even have the ability to do so. But pretty cool nonetheless. And a bit frightening that this is what goes through my head in the wee hours of the morning...
OK, so not so bad after all. We're talking about the sum of an arithmetic series here, so the general formula for the sum of such a series says to average the extreme values of the series and then to multiply by the number of terms in the series. In this particular case, the sum of the arithmetic series that starts with 1 and ends with N is:
ReplyDelete(1 + N)/2 * N
If N is odd, then (1 + N) is even, making (1 + N)/2 an even integer. Multiplying that number by N makes N a factor of the product.
On the other hand, if N is even, then (1 + N) is odd, making (1 + N)/2 a number that ends in .5. Multiplying by N yields an integer product, but alas, because one of the multipliers is not an integer, this product is not a multiple of N.