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### Finding the 1000th Term in a Sequence

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Date: 1/19/96 at 15:34:21
From: Anonymous
Subject: Number

Our class was given the following problem in class: Two kids are on a
car trip, and there's nothing to do, so they decide to count telephone
poles. One kid counts normally, 1,2,3,4,5...25,26,27...31,32,33, etc.
But the other kid counts them a different way. He counts them like this:
1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1, and then up to twenty, and down
again, then thirty, and down, etc. When the normal kid gets to one
thousand, their father tells them to,"SHUT UP!" What number was the
weird kid on when their father told them to shut up, if both kids were
counting at the same time?
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Date: 1/21/96 at 20:54:44
From: Doctor Ken
Subject: Re: Number

Hey, neat question!

Essentially, you're looking for the 1000th term in the sequence
1,2,3,4,5,6,7,8,9,10,9,8,7,6,5,4,3,2,1,2,...29,30,29,28,...., which
looks like it could be pretty hairy.  Let's see if we can find a pattern
or a formula:

1,2,3,4,5,6,7,8,9,10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 2,...29,30,29,28,....,
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...

See what I did?  I lined up the two kids' counting methods next to each
other.  Notice that when the weird kid starts over at 1, the normal kid
hits 19.  That's because the weird kid has 10+9=19 terms in the
first part of his sequence (until he starts over at 1).  The next part
of his sequence will be 20+19=39 terms, for a total of 19+39 = 58 terms.

If you wanted to, you could just keep going like this, adding 19 + 39 +
59 + 79 + ... until you get just below 1000, and then just do it by hand
from there.

Or, if you want to be REALLY slick, you could figure out a formula
for the sum of those series: a formula where you could plug in the
number of groups (a group is a count-up and a count-down), and you'd
get the number of terms in the weird kid's sequence until then.  For
instance, you would plug in 1 and you'd get 19, and you'd plug in 2 and
you'd get 19 + 39 = 58, and so on.

Here are some hints if you wanted to do it that way:

The sequence is also known as
20 + 40 + 60 + ... + 20*n - n.  Unless I messed up.

1+2+3+4+5+6+...+n = n(n+1)/2.

See if this sheds any light on the problem for you.  Good luck!

-Doctor Ken,  The Math Forum

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Associated Topics:
High School Sequences, Series

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