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Sequences

Presented by:

Dana TeCroney

Let S be a sequence of consecutive positive integers whose terms have a sum of 2007.  Find all such sequences S and prove that you have found them all.  (Do not include the trivial sequence of 2007 alone.)

Since this is a sequence of positive integers whoÕs sum is 2007, it will be an arithmetic sum, given by:

where

n is the number of terms

a1 is the first term of the sequence

an is the last term

This equation can be simplified to .  Now, 4014 = 2 * 32 * 223, but what does this mean?

There is a sequence with 2, 2 * 3, 2 * 32, 32, 223, 2 * 223, 3  * 223, and 32 * 223 terms in them.  How many of these fit our criteria though?

LetÕs explore some of these sequences:

n = 2

1003 + 1004 = 2007

One way to think about this is if you took 2007/2 = 1003.5, which means you need one integer (n/2) above 1003.5 and one below.  This method can be used for all even n.

n = 2 * 3

332 + 333 + 334 + 335 + 336 + 337 = 2007

2007/6 = 334.5:  n/2 = 3, so you need three integers above 334.5, and three below.

n = 3

668 + 669 + 670 = 2007

In the case of an odd n, 2007/n will be the middle number of the sequence (2007/3 = 669) and you will have (n-1)/2 terms on each side of the middle number.

2007/223 = 9; so, 9 is our middle number.  You will have 222/2 = 111 terms less than 9 (and greater than 9), but what is wrong with this?  This violates our domain, assuming we have a sequence of positive integers.

It follows from this that none of the sequences with 223, 2 * 223, 3 * 223, or 32 * 223 will adhere the domain.

This means that there will be five sequences (n = 2, 3, 6, 9, 18) of consecutive positive integers that sum to 2007.

n = 9

219 + 220 + 221 + 222 + 223 + 224 + 225 + 226 + 227 = 2007

n = 18

103 + ... + 120 = 2007

Possible Extensions:

What techniques could student come up with to find the number of sequences?  What about the sequences themselves if they were given the number of terms?  If x and y werenÕt restricted to the positive integers, then how many sequences would there be?  What are they?  It would be interesting to see what kind of method students would use if the number were changed to something other than 2007.