An infinite sequence of positive integers is called an "

[sachin_yadav]

An infinite sequence of positive integers is called an "alpha sequence" if the number of even integers in the sequence is finite. If S is an infinite sequence of positive integers, is S an alpha sequence?

(1) The first ten integers in S are even.
(2) An infinite number of integers in S are odd.

Answer is E

Got it, but not sure. So, can anyone please explain in detail, how answer is E ?

Regards
Sachin

[Ian Stewart]

An infinite sequence of positive integers is called an "alpha sequence" if the number of even integers in the sequence is finite. If S is an infinite sequence of positive integers, is S an alpha sequence?

(1) The first ten integers in S are even.
(2) An infinite number of integers in S are odd.

A sequence is just a list of numbers in order. We can create two sequences which satisfy both statements here, one of which will have an infinite number of even values, and one of which will have a finite number of even values. We can just make the first ten values of our sequence all equal to 8, say, just to ensure Statement 1 is true. The remaining values could just be all of the positive integers, in which case we will have an infinite number of even values and an infinite number of odd values:

8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11....

or we can make the remaining values just the odd positive integers only, in which case we will only have ten even values in our sequence:


8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 3, 5, 7, 9, 11, 13, 15, 17, ....

So even with both statements we have no way of knowing if there is an infinite number of even values in the sequence, and the answer is E.

[sachin_yadav]

Thanks Ian

Regards
Sachin

[LalaB]

we are told that an infinite sequence of positive integers is called an "alpha sequence" if the number of even integers in the sequence is finite. to answer to the q. we need 2 things to be true- 1. the sequence is infinite 2. the number of even integers in this sequence is finite.
the 1st one is already told in the q. so ,we need to prove the 2nd one.

(1) it could be enough, if we were told that "The first ten integers in S are even, and no other even numbers exist." since in an original post it was said only about 1st 10 integers, we have no info about other integers.as a result, they can be even or odd. so , (1) is insuff

(2) it could be enough, if we were told that "all numbers of sequence S are odd." since (2) says only about some integers, no conclusion can be made about all integers.

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The only thing you have to be careful is that you do not assume that the sequence is in AP. If you assume that or if it is given in the question, then the answer comes out to be D and not E.

Just changed a scenario a bit and found that the answer turns out to be D...

Hope this was helpful in some ways...