Comparing Sums of Odd and Even Integers
Date: 03/31/2004 at 19:13:20 From: jules Subject: total of odd & even numbers Which is greater, the sum of all of the even numbers that are less than 100, or the sum of all of the odd numbers that are less than 100? It takes too long to add them all up!
Date: 04/01/2004 at 13:52:26 From: Doctor Ian Subject: Re: total of odd & even numbers Hi Jules, It would take a long time to add them all up, wouldn't it? There is a formula you can use to find such sums, but let's skip that and see if we can be clever and just think about the question. Let's look at a smaller version of the same problem, where we go up to 10 instead of 100. We want to compare 0 + 2 + 4 + 6 + 8 against 1 + 3 + 5 + 7 + 9 without actually adding them up. What if we line the numbers up like this? 0, 2, 4, 6, 8 1, 3, 5, 7, 9 In each case, the number on the bottom is larger than the one on top, right? And we have the same number of numbers in each list. So if we add the ones on the bottom, we have to come out with something larger than if we add the ones on top, don't we? Here's another way to look at it. Suppose we start even smaller, with the numbers less than 2: Even: 0 Odd: 1 The sum of the odds is larger, right? Now let's add a couple more numbers: Even: 0, 2 Odd: 1, 3 Since I added a larger number to the odds, the sum of the odds must _still_ be larger, right? And every time I do that, the same thing happens: Even: 0, 2, 4 Odd: 1, 3, 5 5 > 4 Even: 0, 2, 4, 6 Odd: 1, 3, 5, 7 7 > 6 Even: 0, 2, 4, 6, 8 Odd: 1, 3, 5, 7, 9 9 > 8 Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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