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Re: arithmetic progression that's also a geometric one, with a known largest number, need advice
Posted:
Jan 25, 2005 8:22 PM


"Edwill" <nosmapn@lamof.com> wrote in message news:ct6j3b$g3e$1@titan.btinternet.com... > 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 nondecreasing, then the reversed sequence is nonincreasing, so you might as well focus to start with on the nonincreasing 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 nonincreasing 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 nondecreasing sequences that correspond to the nonincreasing ones you've found...
Hope this helps, Mike.



