Making a Series Sum to ZeroDate: 05/24/2002 at 13:01:01 From: Linda Subject: 1,2,3 ........2005 How can I add only + and - signs between 1^2, 2^2, 3^2..... and 2005^2 and make it finally equal zero?? Date: 05/24/2002 at 14:01:45 From: Doctor Douglas Subject: Re: 1,2,3 ........2005 Hi, Linda, This is a very interesting question! First of all, there are 2005 numbers, 1002 of which are even and 1003 of which are odd. The 1003 odd numbers must sum to an odd number, and the 1002 even numbers must sum to an even number, so the final total must be ODD, and cannot be zero. But we can make the final total equal to 1. Let's use the following algebraic identity: (n+1)^2 - n^2 = (n^2 + 2n + 1) - n^2 = 2n + 1 In other words, if you subtract adjacent terms, you will get this progression (let's make sure that 2005^2 is the top of one of the subtractions--this leaves the 1^2 at the bottom by itself): 1 4 9 16 25 36 49 .... 2004^2 2005^2 1 (9-4) (25-16) (49-36) .... (2005^2 - 2004^2) 1 5 9 13 .... 4009 You'll need to put a "-" in front of 4,16,36, etc., and a "+" in front of the other numbers to get this sequence (at least so far). If we end up subtracting the 5 (i.e., 9-4), then we'll have to have a + in front of the original 4 and a - in front of the original 9. (Does that make sense?) Now see if you can figure out what to do with this last sequence of numbers. Note that their differences are all four: 4 = 5 - 1 = 13 - 9 = ... - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ Date: 05/24/2002 at 15:44:12 From: Linda Subject: 1,2,3 ........2005 Dr.Douglas, Thank you so much for your answer!!! |
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