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An infinite sequence of positive integers is called an "alph

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An infinite sequence of positive integers is called an "alph [#permalink] New post 16 Nov 2007, 17:22
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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.
[Reveal] Spoiler: OA

Last edited by Bunuel on 23 Jul 2014, 04:24, edited 3 times in total.
Renamed the topic, edited the question, added the OA and moved to DS forum.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 16 Nov 2007, 20:53
Statement 1 is not sufficient.

It says that the first 10 integers of S are even. That does not imply there are no even integers in the rest of the sequence.

Statement 2 is not sufficient either as it is of no relevance.

Answer is E.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 16 Nov 2007, 22:36
Ev123 wrote:
An infinite sequence of positive integers is called an
Quote:
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 10 integers of S are even

2. An infinite number of integers in S are odd.


OA is E.

What is an Alpha Sequence? Based on the def. I thought it was A.
Can someone please elaborate?

Thanks!
~ Ev


As jbs mentioned, A doesn't tell us whether the number of even ints. are finite or not in the sequence; it only tells us that the 1st ten ints. are even. Therefore, A is not suff.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 26 Sep 2010, 14:49
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)
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 26 Sep 2010, 15:09
shrouded1 wrote:
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)


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.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 26 Sep 2010, 15:16
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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
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 26 Sep 2010, 15:34
shrouded1 wrote:
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


That's what I needed to know! :-D

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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 20 Nov 2010, 18:26
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?
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 21 Nov 2010, 14:09
vicksikand wrote:
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?


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
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 21 Nov 2010, 15:11
shrouded1 wrote:
vicksikand wrote:
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?


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


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.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 22 Nov 2010, 21:16
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
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 23 Nov 2010, 07:38
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?
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 24 Nov 2010, 18:51
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vicksikand wrote:
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?


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)
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 25 Nov 2010, 06:57
I posted my query on a math forum (Dr. Math) to be specific and below is the summary of the response I got.

1. Sequences and Progressions are one and the same.
2. Any function can constitute a sequence, but its not true the other way around. Functions are a broader concept.
3. Any random* group of numbers may constitute a sequence: A sequence does not have to have a well defined formula. The well defined ones are the special cases.
4. A series is the sum of all the terms in a sequence.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 29 Nov 2010, 01:40
vicksikand wrote:
I posted my query on a math forum (Dr. Math) to be specific and below is the summary of the response I got.

1. Sequences and Progressions are one and the same.
2. Any function can constitute a sequence, but its not true the other way around. Functions are a broader concept.
3. Any random* group of numbers may constitute a sequence: A sequence does not have to have a well defined formula. The well defined ones are the special cases.
4. A series is the sum of all the terms in a sequence.

Can someone throw some light on the 4th point mentioned above..I somehow disagree with point 4.

IMO, some of all terms will be single number, not a series..Am I missing something?
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 29 Nov 2010, 05:05
vaibhavtripathi wrote:
vicksikand wrote:
I posted my query on a math forum (Dr. Math) to be specific and below is the summary of the response I got.

1. Sequences and Progressions are one and the same.
2. Any function can constitute a sequence, but its not true the other way around. Functions are a broader concept.
3. Any random* group of numbers may constitute a sequence: A sequence does not have to have a well defined formula. The well defined ones are the special cases.
4. A series is the sum of all the terms in a sequence.

Can someone throw some light on the 4th point mentioned above..I somehow disagree with point 4.

IMO, some of all terms will be single number, not a series..Am I missing something?


Probably I didnt word the sentence carefully:
If a sequence is : 1,2,3,4,5,...100
The corresponding series is : 1+2+3+4+5+....100
And yes, the sum of the terms in a series will be a single number.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 29 Nov 2010, 23:32
A caveat is that for a function to constitute a sequence, the function must be defined over a countable set. The basic idea is a sequence or a progression is an ordered set, i.e., there is a first a second a third and so on element. So the function must define this order.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 30 Nov 2010, 01:23
A sequence doesnt necessarily have to have any order associated with it. The special cases in which we have order are our progressions.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 30 Nov 2010, 04:55
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vicksikand wrote:
A sequence doesnt necessarily have to have any order associated with it. The special cases in which we have order are our progressions.


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.

http://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.
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Re: An infinite sequence of positive integers is called an "alph [#permalink] New post 12 Sep 2013, 07:25
VeritasPrepKarishma wrote:
vicksikand wrote:
A sequence doesnt necessarily have to have any order associated with it. The special cases in which we have order are our progressions.


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.

http://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.


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?
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Re: An infinite sequence of positive integers is called an "alph   [#permalink] 12 Sep 2013, 07:25
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