Z is an infinite sequence of distinct positive real numbers. Is there

Question

Z is an infinite sequence of distinct positive real numbers. Is there a number in sequence Z that is greater than every other number in Z?

(1) No number in sequence Z is greater than 1.

(2) 1 is in sequence Z.

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Source:  Ian Stewart

Akp880 · Answer

Z is an infinite sequence of distinct positive real numbers. Is there a number in sequence Z that is greater than every other number in Z?

Explanation:

(S1) No number in sequence Z is greater than 1.
Not sufficient information to comment .
 S1 insuffcient

(S2) 1 is in sequence Z.
Not sufficient information to comment .
 S2 insuffcient

Combining S1 & S2

given that No number in sequence Z is greater than 1 & also given that 1 is in sequence

Hence 1 will be greater than every other number in Z ( already given Z is an infinite sequence of  distinct positive real numbers )

Thus C is the correct answer.

TestPrepUnlimited · Answer

Think of an infinite sequence as a gigantically large amount of numbers instead of "infinite". An infinite sequence can have a fixed range, such as  f(n) = (0.5)^n  with the input n being the integers 1 to infinity. The sequence starts off at 0.5 and keeps on getting smaller, it hits close to 0 but never actually reaches 0.

Statement 1: 
Think of the example above, it is possible to have all integers greater than 0 yet no element of the sequence is equal to 0. Using this parody, it is possible that all numbers in Z are smaller than 1 but 1 is not within the sequence, therefore we would answer "no". Likewise, we can simply include 1 in Z and we would answer "yes".

Statement 2: 
Insufficient.

Combined: 
We confirmed 1 is in the sequence now so sufficient.

A good example to answer "no" with statement 1 alone is  f(n) = 1 - (0.5)^n , where  n > 0 . When we get to very high values of n,  f(n)  will be close to 1 but we will always find a greater value by inputing the next n. Therefore there is no number within the sequence that is greater than the rest of the elements.

Ans: C

MaryKaaviya · Answer

I think the question by itself answers. If there are distinct real numbers infinitely surely one number will be greater that the rest.

Bunuel  is this a GMAT type question

mcmoorthy · Answer

Agree  MaryKaaviya . If the question asks if there is an integer greater than all other numbers in sequence Z then it makes better sense.

Posted from my mobile device

MaryKaaviya · Answer

VeritasKarishma   TestPrepUnlimited  How is this a question in GMAT terms. In GMAT we should assume all basic concepts.

TestPrepUnlimited · Answer

Hi  MaryKaaviya  ,

I don't think you'll see infinite sequences in GMAT so this question is likely outside of the testing range. This is more of a practice question on infinite number logic and it's good for developing the concept of infinity.

KarishmaB · Answer

Z has DISTINCT positive numbers. It has infinite numbers. This is possible in many ways. 
e.g. 1, 2, 3, 4, ... - No maximum number because it is infinite but it has a minimum value of 1
1/2, 1/4, 1/8, 1/16 ... - No minimum because it extends infinitely but has a maximum value of 1/2
and so on... It needn't be an AP or a GP but these are the first ones that come to mind.

(1) No number in sequence Z is greater than 1.

This means that all numbers are less than or equal to 1. So all numbers lie between 0 and 1 but we don't know whether 1 is included or not. It could be an increasing sequence or a decreasing sequence eg.

Increasing sequence: (1 - 1/2), (1 - 1/4), (1 - 1/8) and so on... (No greatest number)
Decreasing sequence: 1, 1/2, 1/4, 1/8, 1/16 ... (1 is greatest number)
Not sufficient alone.

(2) 1 is in sequence Z.
Not sufficient alone.

Using both, we know that 1 is in the sequence and all numbers lie between 0 and 1 so certainly 1 is greater than all other numbers. 
Answer (C)

KarishmaB · Answer

As for its GMAT relevance, it is a fair question. It is not asking you for the sum of the infinite sequence or some such thing. It is testing you on whether you understand the concept of an infinite sequence. I would like my students to know it.

IanStewart · Answer

You certainly do see infinite sets and sequences on the GMAT -- Q166 in the PS section of QR2018 is a very typical example of how the GMAT tests 'infinity', and there are two questions in a row in the same section of that book (Q137 and Q138) that both explicitly contain infinite sequences (though neither truly tests the concept of 'infinity'). Q137 even begins "The infinite sequence a_1, a_2, ..." so infinite sequences are clearly within the scope of the test, though as Karishma correctly points out, they are strict limits on what they can ask about infinite sequences; they can't ask you to sum an infinite sequence, for example, since you need calculus to prove an infinite sum 'converges' and calculus is beyond the scope of the GMAT.

I'm credited as the source of the question in this thread, but I'm not sure where it comes from. It's not from my books or problem sets, but it's possible I posted it on the forum at one point, either to illustrate a fair way the GMAT can test 'infinity', or when suggesting a way to fix a problem someone else had posted.

A couple of the replies above suggest that no statements are needed to answer the question, and that's not the case. It's possible for an infinite sequence to have no minimum or maximum value which actually belongs to the sequence. For example, if the nth term a_n of a sequence was given by the rule:

a_n = \frac{1}{2} + \left( \frac{n}{2n+1} \right)(-1)^{n}

then you get a sequence where every term is strictly between 0 and 1, and which gets closer to 0 when n is odd, and closer to 1 when n is even, the larger the value of n becomes. This sequence never exceeds 1, but it keeps getting bigger and bigger the larger n is, when n is even, and it never drops below zero, but it gets smaller and smaller the larger n is, when n is odd. Or a different example with no minimum or maximum value strictly in the sequence, but which is always between 0 and 1:

0.9, 0.1, 0.99, 0.01, 0.999, 0.001, 0.9999, 0.0001, ....

That's why both statements are necessary in this question.

Bunuel · Answer

IanStewart , this question comes from one of your posts in another forum where you were commenting on a similar but flawed question.

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Z is an infinite sequence of distinct positive real numbers. Is there

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