Arithmetic SeriesDate: 5/19/96 at 13:43:38 From: Ian Balchin Subject: Math problem! Hello. I was helping my son do a project recently. We had a "chessboard" of varying size, 4x4, 5x5....8x8 and so on. We then had an ell of varying size, say 3x2, and had to derive expressions for the sum of the numbers covered by this when placed on the chessboard. We worked up varying solutions but a general solution for any sized board combined with any sized ell in any orientation eluded us. This was because neither of us knew how to sum a series of numbers given the first number, the number of them and the rate of progression, i.e. how could we sum a given number of terms knowing how to calculate the series, i.e. 2,4,6,8... for say 3 terms starting anywhere in the series _not by adding 3 specific terms together_ but by using the first term and the number 3? We also wanted to do this for 1,2,3,4, or any other arithmetic series. If you can help, this would be greatly appreciated. Thanks. Regards, Ian. Date: 5/20/96 at 13:5:47 From: Doctor Ken Subject: Re: Math problem! Hello! I'm not sure I understand exactly what the problem is and how the chessboards and ells work, but I can help you sum arithmetic series. Look at this sum: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 Now, how can we sum this in a clever way? Well, notice that we can pair the 1 and the 9 to get 10, and the 2 and the 8, and so on. So how many 10's will we have? Counting, we see that we have 4 pairs and 1 half-pair. So we can think of this as 4 and 1/2 pairs that are each worth 10, so the total would be 9 x 10 / 2. This formula works in general: the sum of the integers from 1 to n is n(n+1)/2. Can you convince yourself that it works if n is even too? Now see if you can come up with a more general formula for the sum of an arithmetic series based on this one. Good luck! -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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