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.

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) (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 is a positive integer,
and is an integer term of an infinite geometric sequence with common ratio ,
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 terms that are integers.

If , regardless of the value of first term ,
, and all subsequent terms will be 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 terms that are integers either.
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