_______________________________________________________________________________________________________

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

a_{1}
is the first term of the sequence

a_{n}
is the last term

This
equation can be simplified to . Now, 4014 = 2
* 3^{2} * 223, but what does this mean?

There
is a sequence with 2, 2 * 3, 2 * 3^{2}, 3^{2}, 223, 2 * 223,
3 * 223, and 3^{2} * 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.

What
about 223 terms?

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 3^{2}
* 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.