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Arithmetic Series

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Date: 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.
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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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Associated Topics:
High School Sequences, Series

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