An infinite sequence of positive integers is called an "alph

Question

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.

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

KarishmaB · Accepted Answer

Sequence means a succession of numbers whose order is determined by a rule or a function. Even 0, 3, 0 , 3, 0, 3 ... is a sequence. It has a rule defining it.

A progression is a simple sequence of numbers in which there is a constant relation between consecutive terms.

A series is the sum of the terms of a sequence.

All progressions are sequences but all sequences are not progressions. (though the terms are used interchangeably often)

shrouded1 · Answer

1) not sufficient. As it gives no information about the terms after the first ten

2) not sufficient. Eg 1) all odd positive integers 2) all positive integers of the form 4k and 4k+1. first one is an alpha sequence 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

Answer (e)

metallicafan · Answer

Yes, I thought the same, because there are no restrictions. However, I'm trying to imagine a sequence in which the first 10 integers are even, and the rest are odd and even, or only odd. In other words, I'm trying to imagine the function of those sequences, and I cannot find it :s
That information would be helpful to be more sure about our answer.

shrouded1 · Answer

Sequences need not have a neat and clean functional form, and this one might look messy. But here is some examples :

First ten even numbers followed by all odd numbers > 99

First ten even numbers next ten odd numbers next ten even numbers and so on ...

The sequence is not an ap or gp so can be quite arbitrary

vicksikand · Answer

If its a sequence be it an Arithmetic, Geometric or Harmonic progression - it has to have some order. 
Given S is an infinite Sequence of positive Integers.
1: 1st 10 in S are even - makes you wonder whether the sequence has infinite even's.
2,4,6,8,10,12....2k
or it could be 4,10,16,...... a+6
I really couldn't find a way to introduce Odd numbers after the 1st 10 Evens. Unless there are special cases such as:
f=2k for 1<k<11 and 
f=2k+1 for k>11. But then this wouldnt be a sequence , right?
2: Infin number of integers in S are odd : Not sufficient.

Any views?

shrouded1 · Answer

A sequence of integers doesnt need to be an AP, GP, or HP necessarily. All you need is a well defined set of rules, in order to get a sequence. For instance the following is an example :

Sequence S, such that the ith element is given by :
 s_i = 2i  for i=1 to 10
 s_i=4i+3  for i>10

So this is an infinite sequence in which only the first 10 numbers are even and the rest are an infinite number of odd numbers

vicksikand · Answer

s_i = 2i  for i=1 to 10
 s_i=4i+3  for i>10

This sounds more like a function than a sequence and the values of i for which the function is valid - formulate the domain.

Correct me if I am wrong:
In mathematical terms a progression and a sequence are one and the same. 
The only series/progressions I am aware of are : AP,GP,HP,Fibonacci, Cauchy, and Farey.

shrouded1 · Answer

progressions and sequences are subsets of functions such that the domain is the set of integers.

For an AP for instance we know  s_n = a + (n-1)d

AP,GP,HP,Fibonacci, Cauchy etc etc are just names for special cases of sequences, they are by no means exhaustive ... any ordered set of numbers can qualify as a sequence
So even {1,-1,1,-1,1,-1,1,-1,.....} is a perfectly valid sequence

vicksikand · Answer

A progression may very well a subset of a sequence ( arithmetic progression = arithmetic sequence), and yes a progression is the subset of a function. 
What constitutes a progression? All progressions = Sequences , is it true the other way around?
Can the range of a function be defined as a sequence?

vicksikand · Answer

A sequence doesnt necessarily have to have any order associated with it. The special cases in which we have order are our progressions.

KarishmaB · Answer

Sequence, in mathematics, is an ordered set of mathematical quantities called terms.
There is always an order associated with a sequence. A collection of terms without any order is called a Set.
Check out this link for explanation.

https://encyclopedia2.thefreedictionary.com/Sequence+%28mathematics%29

Anyway, I don't think we are going anywhere with this discussion and are wasting far too much time. You will not be asked to define Progressions/Sequences/Series in GMAT and I doubt you will get a question where differentiating between them will be needed.

obs23 · Answer

I was just wondering, when, if at all, I would know that a sequence is finite please? Do I understand correctly that the only way i would know this is if GMAT says it in a convoluted manner... or is there a way to know in advance what sequence may be finite?

Bunuel · Answer

If it is a relevant info, then this would be clear from the stem or the statements. For more: Theory on sequences problems: sequences-progressions-101891.html DS sequences problems to practice: search.php?search_id=tag&tag_id=111 PS sequences problems to practice: search.php?search_id=tag&tag_id=112

ADisHere · Answer

broke it down in this easy approach

S should have finite evens

st 1- first 10 are even, but what about the others? reject
s 2- infinite odds, but do we know about evens? no reject

Combining both - we still don't have any surety whether there are finite evens
Hence we will go with E.

agrasan · Answer

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.

(1) The first ten integers in S are even.
 Insufficient , it is possible that integers from 50th term to n (very large) are even

(2) An infinite number of integers in S are odd.
 Insufficient , it doesn't tell us anything about number of even numbers

Combining (1) and (2),
 Insufficient , it is possible that even integers are also infinite in number so (C) is not certain.
There might be a pitfall here that  infinite + small number = infinite  but that's not true.

Hence,  (E)  is the answer.

An infinite sequence of positive integers is called an "alph

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An infinite sequence of positive integers is called an "alph

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