Question: An infinite sequence of positive integers is called as “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

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Approach Solution (1):
S1 is not sufficient as it gives no information about the terms after the first ten
S2 is also not sufficient.
1: All odd positive integer
2. All positive integers of the form 4k and 4k + 1
First one is an alpha sequence and second isn’t
1 + 2 not sufficient
Easy to see this as making the first ten terms even or odd doesn’t put any restrictions on the rest.
We can form sequences similar to the above
An infinite sequence may have infinite even as well as infinite odd terms
Correct option: E

Approach Solution (2):
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 are an infinite number of even values in the sequence.
Correct option: E

Approach Solution (3):
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 insufficient.
(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.
Correct option: E

“An infinite sequence of positive integers is called as “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?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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