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Math Expert V
Joined: 02 Sep 2009
Posts: 56357

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40 00:00

Difficulty:   95% (hard)

Question Stats: 48% (02:18) correct 52% (02:11) wrong based on 309 sessions

### HideShow timer Statistics The sequence $$a_1$$, $$a_2$$, $$a_3$$, ..., $$a_n$$, ... is such that $$i*a_i=j*a_j$$ for any pair of positive integers $$(i, j)$$. If $$a_1$$ is a positive integer, which of the following could be true?

I. $$2*a_{100}=a_{99}+a_{98}$$

II. $$a_1$$ is the only integer in the sequence

III. The sequence does not contain negative numbers

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III

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Math Expert V
Joined: 02 Sep 2009
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5
Official Solution:

The sequence $$a_1$$, $$a_2$$, $$a_3$$, ..., $$a_n$$, ... is such that $$i*a_i=j*a_j$$ for any pair of positive integers $$(i, j)$$. If $$a_1$$ is a positive integer, which of the following could be true?

I. $$2*a_{100}=a_{99}+a_{98}$$

II. $$a_1$$ is the only integer in the sequence

III. The sequence does not contain negative numbers

A. I only
B. II only
C. I and III only
D. II and III only
E. I, II, and III

Given that the sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$, ... have the following properties: $$i*a_i=j*a_j$$ and $$a_1=\text{positive integer}$$, so $$1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=\text{positive integer}$$.

We should determine whether the options given below can occur (notice that the question is which of the following COULD be true, not MUS be true).

I. $$2a_{100}=a_{99}+a_{98}$$. Since $$100a_{100}=99a_{99}=98a_{98}$$, then $$2a_{100}=\frac{100}{99}a_{100}+\frac{100}{98}a_{100}$$. Reduce by $$a_{100}$$: $$2=\frac{100}{99}+\frac{100}{98}$$ which is not true. Hence this option could NOT be true.

II. $$a_1$$ is the only integer in the sequence. If $$a_1=1$$, then all other terms will be non-integers, because in this case we would have $$a_1=1=2a_2=3a_3=...$$, which leads to $$a_2=\frac{1}{2}$$, $$a_3=\frac{1}{3}$$, $$a_4=\frac{1}{4}$$, and so on. Hence this option could be true.

III. The sequence does not contain negative numbers. Since given that $$a_1=\text{positive integer}=n*a_n$$, then $$a_n=\frac{\text{positive integer}}{n}=\text{positive number}$$, hence this option is always true.

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1
Hi Bunuel,

I got this question on a GMAT Club test and even after reviewing it I'm having trouble understanding what is going on. The following deduction is giving me trouble:

since 100a(sub)100= 99a(sub)99 =98a(sub)98, then 2a(sub)100 = [99/]a(sub)100 + [98/]a(sub)100. <<------- Please see attachment below if this is not clear.

I hope this makes sense. Cheers.
>> !!!

You do not have the required permissions to view the files attached to this post.

Math Expert V
Joined: 02 Sep 2009
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Darknightw wrote:
Hi Bunuel,

I got this question on a GMAT Club test and even after reviewing it I'm having trouble understanding what is going on. The following deduction is giving me trouble:

since 100a(sub)100= 99a(sub)99 =98a(sub)98, then 2a(sub)100 = [99/]a(sub)100 + [98/]a(sub)100. <<------- Please see attachment below if this is not clear.

I hope this makes sense. Cheers.

From $$100a_{100}=99a_{99}$$ --> $$a_{99}=\frac{100}{99}a_{100}$$;

From $$100a_{100}=98a_{98}$$ --> $$a_{98}=\frac{100}{98}a_{100}$$;

So, option I. $$2a_{100}=a_{99}+a_{98}$$ becomes: $$2a_{100}=\frac{100}{99}a_{100}+\frac{100}{98}a_{100}$$.

Hope it's clear.
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Bunuel,

Thanks for the quick response. Yes, it's now very clear. Can't believe I couldn't see that!

Thanks again!

Darknightw

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Senior Manager  Joined: 31 Mar 2016
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GMAT 1: 670 Q48 V34 GPA: 3.8
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I think this is a high-quality question and I agree with explanation.
Intern  Joined: 22 Jul 2013
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Location: United States
Concentration: Technology, Entrepreneurship
Schools: IIM A '15
GMAT 1: 650 Q46 V34 GMAT 2: 720 Q49 V38 GPA: 3.67
WE: Engineering (Non-Profit and Government)

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Hi Bunuel,

Quick question. I solved the question as below. I assume that a(x) as a function, which reciprocates the integer x. so 100a100 = 100(1/100) = 1. Thus even negative integer say -1a(-1) will give a value of 1 and will equal any other xa(x) value.
Now I see that Stmt 1: 2 (1/100) = (1/99)+(1/98) Not Possible
Stmt 2: a1 = 1/1 = 1; every thing else is 1/2; 1/-1 ( still an integer) meaning no negative numbers in the sequence
Stmt 3: As above the sequence may or may not contain negative numbers. Since the question stem says " Could be true" I chose Stmt 2 and 3 to be the right answer.

Is my reasoning valid?

Thanks,
Arun
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Schools: UConn"19
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confusing question
Senior Manager  S
Joined: 08 Jun 2015
Posts: 421
Location: India
GMAT 1: 640 Q48 V29 GMAT 2: 700 Q48 V38 GPA: 3.33

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The answer must be option D. There are some awesome explanations above. I did it using a slightly different method though.

ai/aj=j/i

express a100 , a99, & a98 in terms of a1.

Now go to each and every statement.

Statement 1 - substitute in LHS,RHS. Solve and you get LHS not equal to RHS. Hence 1 is wrong
Since 1 is wrong we are left with choices B and D. II is common to both. Jump on to statement III.

III- a1 is positive i & j are positive so any number of the form an will be positive. Hence III is true.

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Can anyone please provide an alternative solution of this problem. I have trouble understanding the official answer explanation.

Thanks !
Math Expert V
Joined: 02 Sep 2009
Posts: 56357

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jyotipes21@gmail.com wrote:
Can anyone please provide an alternative solution of this problem. I have trouble understanding the official answer explanation.

Thanks !

Check here: https://gmatclub.com/forum/series-a-n-i ... 27175.html
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The word "could be" makes the difference. Otherwise, option II is not always correct. For instance, if a1 = 2 , there could be two integers in the sequence.
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GMAT 1: 630 Q43 V34 ### Show Tags

1
That's the problem!!
>> !!!

You do not have the required permissions to view the files attached to this post.

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Math Expert V
Joined: 02 Sep 2009
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patto wrote:
That's the problem!!

Thank you. We are looking into this. Could you see posts in this discussion OK? For example, could you see this post: https://gmatclub.com/forum/m17-184131.html#p1453637
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Bunuel wrote:
patto wrote:
That's the problem!!

Thank you. We are looking into this. Could you see posts in this discussion OK? For example, could you see this post: https://gmatclub.com/forum/m17-184131.html#p1453637

Yes i can see all the posts but in all of them, the numbers are one in front the other in the same place.
I found this problem in all the discussions that correspond to the questions of the same test(my first test ?)

Posted from my mobile device
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Math Expert V
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patto wrote:
Bunuel wrote:
patto wrote:
That's the problem!!

Thank you. We are looking into this. Could you see posts in this discussion OK? For example, could you see this post: https://gmatclub.com/forum/m17-184131.html#p1453637

Yes i can see all the posts but in all of them, the numbers are one in front the other in the same place.
I found this problem in all the discussions that correspond to the questions of the same test(my first test ?)

Posted from my mobile device

Can you please try now? We fixed it. You might need to clean the cache first.
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Manager  G
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Now It's working fine! Ty Bunuel!!!
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Bunuel wrote:
Official Solution:

II. $$a_1$$ is the only integer in the sequence. If $$a_1=1$$, then all other terms will be non-integers, because in this case we would have $$a_1=1=2a_2=3a_3=...$$, which leads to $$a_2=\frac{1}{2}$$, $$a_3=\frac{1}{3}$$, $$a_4=\frac{1}{4}$$, and so on. Hence this option could be true.

III. The sequence does not contain negative numbers. Since given that $$a_1=\text{positive integer}=n*a_n$$, then $$a_n=\frac{\text{positive integer}}{n}=\text{positive number}$$, hence this option is always true.

Why does the explanation for II say "could be true" and not "always true" (like it does for III)?
Math Expert V
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aserghe1 wrote:
Bunuel wrote:
Official Solution:

II. $$a_1$$ is the only integer in the sequence. If $$a_1=1$$, then all other terms will be non-integers, because in this case we would have $$a_1=1=2a_2=3a_3=...$$, which leads to $$a_2=\frac{1}{2}$$, $$a_3=\frac{1}{3}$$, $$a_4=\frac{1}{4}$$, and so on. Hence this option could be true.

III. The sequence does not contain negative numbers. Since given that $$a_1=\text{positive integer}=n*a_n$$, then $$a_n=\frac{\text{positive integer}}{n}=\text{positive number}$$, hence this option is always true.

Why does the explanation for II say "could be true" and not "always true" (like it does for III)?

The question asks: which of the following COULD be true. If an option is true, is possible, even for one sequence then it fits.

I is not true for this sequence at all. So it's out.
II COULD be true in certain case, so it fits.
III is ALWAYS true, so it also fits.

Hope it's clear.
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For the statment 2, instead of taking a1 as 1 if we take it as 2 then there woudl be 2 integers in this case. Hence the statement can be false aswell
a2 = a1/2 => 2/2 => 1

So why option 2 is correct? Re: M17-25   [#permalink] 03 Aug 2018, 14:52

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# M17-25

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