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Topic: arithmetic progression that's also a geometric one, with a known largest number, need advice
Replies: 4   Last Post: Dec 5, 2012 1:11 AM

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Mike Terry

Posts: 767
Registered: 12/6/04
Re: arithmetic progression that's also a geometric one, with a known largest number, need advice
Posted: Jan 25, 2005 8:22 PM
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"Edwill" <> wrote in message
> Hi all,
> The problem is that an arithmetic progression (a_1,...,a_n), is also a
> geometric progression (after applying some permutation). They (a_i's) are
> all different, and the largest number of them is 2004.
> Any idea to begin with to make a breakthrough?
> So far,
> 1). they have the same sum
> 2). the series is either monotonic increasing, or decreasing
> 3). if 2). is the case, and without loss of generality, can i assume
> a_n=2004. or am i already lost in 2).?
> Thank you very much...I just need some enlightening hint...:D

If the sequence is non-decreasing, then the reversed sequence is
non-increasing, so you might as well focus to start with on the
non-increasing kind - so in this case a_1 = 2004... Then you have to ask
yourself what can n be? If n=1 or n=2, then all non-increasing sequences
are both arithmetic and geometric, and so are possible solutions...

So then you have to think of the case n>2. You can write the general form
for the first three terms, assuming the sequence is arithmetic, and assuming
it is geometric. You should then see the possible solution sequences this
constraint allows...

Finally don't forget the non-decreasing sequences that correspond to the
non-increasing ones you've found...

Hope this helps,

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