SOLUTION: Determine all nonnegative integers r such that it is possible for an infinite geometric sequence to contain exactly r terms that are integers. Prove your answer.

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Question 1116978: Determine all nonnegative integers r such that it is possible for an infinite geometric sequence to contain exactly r terms that are integers. Prove your answer.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I do not believe ther is such a nonnegative integer, but if you know of a different answer, please enlighten me.
A nonnegative integer is either a positive integer or it is zero.
If r is a positive integer,
and a%5Bn%5D is an integer term of an infinite geometric sequence with common ratio r ,
a%5Bn%2B1%5D=a%5Bn%5D%2Ar will be an integer,
and so will be every term after that.
As a consequence, there will be infinite terms that are integers.
In that case, it will not be possible for that infinite geometric sequence to contain exactly r terms that are integers.

If r=0 , regardless of the value of first term a%5B1%5D ,
a%5B2%5D=a%5B1%5D%2A0=0 , and all subsequent terms will be 0 too.
In that case, there will also be an infinite number of terms that are integers,
and that infinite geometric sequence will not contain exactly r terms that are integers either.