Arithmetic Series

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.

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/