Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 03 Feb 2010
Posts: 57

The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
30 Jun 2010, 09:38
Question Stats:
45% (02:13) correct 55% (02:20) wrong based on 168 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 M1725 Series is such that for any pair of positive integers . If is a positive integer, which of the following is possible? I. II. is the only integer in the series III. The series 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
Official Answer and Stats are available only to registered users. Register/ Login.




Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10263
Location: Pune, India

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
07 Feb 2012, 08:41
vinnik wrote: Series 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 is possible?
I. 2*A(100) = A(99) + A(98) II. A(1) is the only integer in the series III. The series does not contain negative numbers
A) I only B) II only C) I & III only D) II & III only E) I, II & III
Will appreciate if anyone explains this question with an easy method.
Thanks & Regards Vinni First thing I want to understand is this relation: i*A(i) = j*A(j) for any pair of positive integers. I will take examples to understand it. When i = 1 and j = 2, A(1) = 2*A(2) So A(2) = A(1)/2 When i = 1 and j = 3, A(1) = 3*A(3) So A(3) = A(1)/3 I see it now. The series is: A(1), A(1)/2, A(1)/3, A(1)/4 and so on... II and III are easily possible. We can see that without any calculations. II. A(1) is the only integer in the series If A(1) = 1, then series becomes 1, 1/2, 1/3, 1/4 ... all fractions except A(1) III. The series does not contain negative numbers Again, same series as above applies. In fact, since A(1) is a positive integer, this must be true. I. 2*A(100) = A(99) + A(98) 2*A(1)/100 = A(1)/99 + A(1)/98 (cancel A(1) from both sides) 2/100 = 1/99 + 1/98 Not true hence this is not possible Answer (D)
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Math Expert
Joined: 02 Sep 2009
Posts: 62676

The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
07 Jul 2010, 16:49
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 nonintegers, 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. Answer: D Hope it's clear.
_________________




Manager
Joined: 05 Jul 2008
Posts: 104

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
30 Jun 2010, 21:56
i*A(i)=j*A(j) for any pair of positive integers means i*A(i) is constant for any i So i * A(i) = C (C is a constant) A(i)= C/i for any positive integers A(1)=C is an integer A(100)= C/100 A(98)= C/98 A(99)=C/99 I. 2 * A(100)= 2C/100= C/50 A(99)=C/99, A(98)=C/98 Because C is an positive integer so C can not be Zero. I is impossible because C/50 cannot equal C(1/99+1/98) II. if C=1, A(n)=C/n so A(1) is the only integer III. C is a positive integer so A(i)= C/i can not be negative D is my answer. By the way, I need 3 more kudos. So if I'm right, Kudo me please



Senior Manager
Joined: 23 Oct 2010
Posts: 314
Location: Azerbaijan
Concentration: Finance

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
07 Feb 2012, 03:34
Bunuel wrote: A sequence of numbers \(a_1\), \(a_2\), \(a_3\), ... have the following properties: \(i*a_i=j*a_j\) and \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\).
Bunuel, I didnt get this part. I seem to misunderstood the q.stem could u please clarify it?
_________________
Happy are those who dream dreams and are ready to pay the price to make them come true
I am still on all gmat forums. msg me if you want to ask me smth



Math Expert
Joined: 02 Sep 2009
Posts: 62676

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
07 Feb 2012, 03:39
LalaB wrote: Bunuel wrote: A sequence of numbers \(a_1\), \(a_2\), \(a_3\), ... have the following properties: \(i*a_i=j*a_j\) and \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\).
Bunuel, I didnt get this part. I seem to misunderstood the q.stem could u please clarify it? Sure. Given: \(a_1=positive \ integer\). Next, \(i*a_i=j*a_j\), notice that we have the same multiple and the same index of a on both sides: \(1*a_1=2*a_2\), \(2*a_2=3*a_3\), \(a_3=4*a_4\).... Hence, \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\) (it equal to an integer since \(a_1=positive \ integer\)). Hope it's clear.
_________________



Senior Manager
Joined: 23 Oct 2010
Posts: 314
Location: Azerbaijan
Concentration: Finance

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
07 Feb 2012, 10:59
got it at last:) thnx
_________________
Happy are those who dream dreams and are ready to pay the price to make them come true
I am still on all gmat forums. msg me if you want to ask me smth



Manager
Joined: 14 Dec 2011
Posts: 71

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
08 Feb 2012, 07:28
Thanks Bunuel and Karishma for clearing my doubt. Regards Vinni



Manager
Joined: 15 Jan 2011
Posts: 88

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
21 May 2012, 07:21
Bunuel could you please go thru this part one more time? Cant get it
Attachments
Screen Shot 20120521 at 7.19.49 PM.png [ 8.49 KiB  Viewed 6648 times ]



Math Expert
Joined: 02 Sep 2009
Posts: 62676

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
21 May 2012, 08:15
Galiya wrote: Bunuel could you please go thru this part one more time? Cant get it 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.
_________________



Manager
Joined: 16 Feb 2011
Posts: 165
Schools: ABCD

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
13 Jul 2012, 11:22
Bunuel, I agree with (i) and (iii). However, I am not sure about (ii).
Why did you substitute a1 =1 ? If A(1) is the only integer => n=1; But how do we know that a1 = 1? a1 could be anything....a1=2 also holds good because there is only one number. Correct? Essentially, there is no A(2), A(3) etc.
Thoughts?



Math Expert
Joined: 02 Sep 2009
Posts: 62676

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
13 Jul 2012, 11:28
voodoochild wrote: Bunuel, I agree with (i) and (iii). However, I am not sure about (ii).
Why did you substitute a1 =1 ? If A(1) is the only integer => n=1; But how do we know that a1 = 1? a1 could be anything....a1=2 also holds good because there is only one number. Correct? Essentially, there is no A(2), A(3) etc.
Thoughts? The question asks "which of the following is possible" or which of the following COULD be true. So, we don't know that \(a_1=1\), but \(a_1\) COULD be 1 and in this case it would be the only integer in the sequence. So, II is certainly POSSIBLE. Hope it's clear.
_________________



Intern
Status: K... M. G...
Joined: 22 Oct 2012
Posts: 26
Concentration: General Management, Leadership
GMAT Date: 08272013
GPA: 3.8

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
09 Feb 2013, 05:15
Bunuel wrote: RuslanMRF wrote: Series 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 is possible?
I. 2A(100)=A(99)+A(98) II. A(1) is the only integer in the series III. The series does not contain negative numbers
I only II only I and III only II and III only I, II, and III
Please, explain the the solution. New edition of this question with a 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 \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\). 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 be true. II. \(a_1\) is the only integer in the sequence. If \(a_1=1\), then all other terms will be nonintegers > \(a_1=1=2a_2=3a_3=...\) > \(a_2=\frac{1}{2}\), \(a_3=\frac{1}{3}\), \(a_4=\frac{1}{4}\), and so on. Hence this option could be true. III. The sequence does not contain negative numbers > since given that \(a_1=positive \ integer=n*a_n\), then \(a_n=\frac{positive \ integer}{n}=positive \ number\), hence this option is always true. Answer: D. Hope it's clear. can you please explain me option A. i am totally confused with it



Math Expert
Joined: 02 Sep 2009
Posts: 62676

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
10 Feb 2013, 02:19
FTG wrote: Bunuel wrote: RuslanMRF wrote: Series 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 is possible?
I. 2A(100)=A(99)+A(98) II. A(1) is the only integer in the series III. The series does not contain negative numbers
I only II only I and III only II and III only I, II, and III
Please, explain the the solution. New edition of this question with a 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 \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\). 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 be true. II. \(a_1\) is the only integer in the sequence. If \(a_1=1\), then all other terms will be nonintegers > \(a_1=1=2a_2=3a_3=...\) > \(a_2=\frac{1}{2}\), \(a_3=\frac{1}{3}\), \(a_4=\frac{1}{4}\), and so on. Hence this option could be true. III. The sequence does not contain negative numbers > since given that \(a_1=positive \ integer=n*a_n\), then \(a_n=\frac{positive \ integer}{n}=positive \ number\), hence this option is always true. Answer: D. Hope it's clear. can you please explain me option A. i am totally confused with it 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.
_________________



Senior Manager
Joined: 07 Apr 2012
Posts: 316

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
22 Sep 2013, 13:25
Bunuel wrote: ksharma12 wrote: Series is such that for any pair of positive integers . If is a positive integer, which of the following is possible? I. II. is the only integer in the series III. The series does not contain negative numbers I only II only I and III only II and III only I, II, and III I have no idea whats going on here? Detailed explanation is appreciated. A set of numbers \(a_1\), \(a_2\), \(a_3\), ... have the following properties: \(i*a_i=j*a_j\) and \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\). We should determine whether the options given below can occur (note that the question is which can be true, not must be true). I. \(2a_{100}=a_{99}+a_{98}\) > as \(100a_{100}=99a_{99}=98a_{98}\), 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 cannot be true. Hi I did not follow the move in bold. Can someone pls. explain a little more?



Math Expert
Joined: 02 Sep 2009
Posts: 62676

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
22 Sep 2013, 23:30
ronr34 wrote: Bunuel wrote: ksharma12 wrote: Series is such that for any pair of positive integers . If is a positive integer, which of the following is possible? I. II. is the only integer in the series III. The series does not contain negative numbers I only II only I and III only II and III only I, II, and III I have no idea whats going on here? Detailed explanation is appreciated. A set of numbers \(a_1\), \(a_2\), \(a_3\), ... have the following properties: \(i*a_i=j*a_j\) and \(a_1=positive \ integer\), so \(1*a_1=2*a_2=3*a_3=4*a_4=5*a_5=...=positive \ integer\). We should determine whether the options given below can occur (note that the question is which can be true, not must be true). I. \(2a_{100}=a_{99}+a_{98}\) > as \(100a_{100}=99a_{99}=98a_{98}\), 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 cannot be true. Hi I did not follow the move in bold. Can someone pls. explain a little more? 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.
_________________



Intern
Joined: 26 Feb 2015
Posts: 25
Location: Thailand
Concentration: Entrepreneurship, Strategy
GMAT 1: 630 Q49 V27 GMAT 2: 680 Q48 V34
GPA: 2.92
WE: Supply Chain Management (Manufacturing)

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
27 Nov 2015, 23:22
Bunuel wrote: II. \(a_1\) is the only integer in the series. If \(a_1=1\), then all other terms will be nonintegers > \(a_1=1=2a_2=3a_3=...\) > \(a_2=\frac{1}{2}\), \(a_3=\frac{1}{3}\), \(a_4=\frac{1}{4}\), and so on. Hence this option can be true.
I don't understand this part. How could I know that A(1) = 1 The question stem mentioned only that "A(1) is a positive integer" If A(1) = 2, then > 1*A(1) = 2*A(2) > A(2) = 1 Then II cannot be true. Please tell me if I get something wrong. Thanks



Math Expert
Joined: 02 Sep 2009
Posts: 62676

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
28 Nov 2015, 07:29
pakasaip wrote: Bunuel wrote: II. \(a_1\) is the only integer in the series. If \(a_1=1\), then all other terms will be nonintegers > \(a_1=1=2a_2=3a_3=...\) > \(a_2=\frac{1}{2}\), \(a_3=\frac{1}{3}\), \(a_4=\frac{1}{4}\), and so on. Hence this option can be true.
I don't understand this part. How could I know that A(1) = 1 The question stem mentioned only that "A(1) is a positive integer" If A(1) = 2, then > 1*A(1) = 2*A(2) > A(2) = 1 Then II cannot be true. Please tell me if I get something wrong. Thanks Please notice that it says "IF \(a_1=1\), ..." and also that the question asks which of the following is possible, so which of the following could be true.
_________________



Current Student
Joined: 03 Apr 2013
Posts: 258
Location: India
Concentration: Marketing, Finance
GMAT 1: 740 Q50 V41
GPA: 3

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
30 Jun 2016, 02:37
vinnik wrote: Series 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 is possible?
I. 2*A(100) = A(99) + A(98) II. A(1) is the only integer in the series III. The series does not contain negative numbers
A) I only B) II only C) I & III only D) II & III only E) I, II & III
Will appreciate if anyone explains this question with an easy method.
Thanks & Regards Vinni Right..only looks complicated(that's how you should think first for some confidence ) Let's give this a shot..
From the given equation we know that.. \(1*A(1) = k*A(k)\) ..where k is any positive integer..
Coming over to the premises..we'll deal with I at the end
II Possible..what if A(1)=1? every other term will be a fraction..so YES
III Always true..no explanation needed
I because it's a "could be true" question..we won't give A(1) a value for this statement..and go with A(1) as..some number/fraction A(1) \(2*A(100) = A(99) + A(98)\)
\(A(100) + A(100) = A(99) + A(98)\) We know that.. \(1*A(1) = 100*A(100)\) \(1*A(1) = 99*A(99)\) and \(1*A(1) = 98*A(98)\)
Using the expressions and transforming the main equation..
\(2*\frac{A(1)}{100} = \frac{A(1)}{99} + \frac{A(1)}{98}\)
\(\frac{A(1)}{50} = \frac{A(1)}{99} + \frac{A(1)}{98}\)
And we know that R.H.S. has no "5" in it...so this will NEVER be true.. Answer (D)



NonHuman User
Joined: 09 Sep 2013
Posts: 14509

Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
Show Tags
29 Feb 2020, 02:17
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________




Re: The sequence a1, a2, a3, ..., an, ... is such that i*a_i=j*aj for any
[#permalink]
29 Feb 2020, 02:17






