Question

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 1

Let, the geometric sequence is such that, value of common ratio is less than 1.

The Sequence is

`6^(n-1),6^(n-2),6^(n-3),.....,.......\infinity.`

The Geometric Squence is infinite geometric sequence, as there are uncountable terms in the sequence.

⇒So, From
`6^(n-1)`, to infinity, there will be n terms which will be integers when , n≥1.

⇒Put, n=1,

Number of terms which are Positive Integers =1 which is
`6^(n-1)`.

⇒When, n=2

Number of terms which are Positive Integers =2 which is
`6^(n-1),6^(n-2)`.

⇒When, n=3

Number of terms which are Positive Integers =3, which is
`6^(n-1),6^(n-2),6^(n-3)`.

..........

So,⇒ when , n=r

Number of terms which are Positive Integers =r, which is
`6^(n-1),6^(n-2),6^(n-3),6^(n-4),........6^(n-r)`.