February 17, 2019 February 17, 2019 07:00 AM PST 09:00 AM PST Attend this Free Algebra Webinar and learn how to master Inequalities and Absolute Value problems on GMAT. February 16, 2019 February 16, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.
Author 
Message 
TAGS:

Hide Tags

Retired Moderator
Status: Enjoying the GMAT journey....
Joined: 26 Aug 2011
Posts: 585
Location: India

A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
29 Mar 2012, 10:51
Question Stats:
33% (02:17) correct 67% (02:05) wrong based on 576 sessions
HideShow timer Statistics
A Farey sequence of order n is the sequence of fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. For example, the Farey sequence of order 3 is: {0, 1/3 , 1/2, 2/3 , 1}. Is sequence S a Farey sequence? (1) Sequence S has fewer than 10 elements. (2) The second element of sequence S is 1/5
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Fire the final bullet only when you are constantly hitting the Bull's eye, till then KEEP PRACTICING.
A WAY TO INCREASE FROM QUANT 3540 TO 47 : http://gmatclub.com/forum/awaytoincreasefromq3540toq138750.html
Q 47/48 To Q 50 + http://gmatclub.com/forum/thefinalclimbquestforq50fromq47129441.html#p1064367
Three good RC strategies http://gmatclub.com/forum/threedifferentstrategiesforattackingrc127287.html




Math Expert
Joined: 02 Sep 2009
Posts: 52902

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
29 Mar 2012, 11:32
A Farey sequence of order n is the sequence of fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. For example, the Farey sequence of order 3 is: {0, 1/3 , 1/2, 2/3 , 1}. Is sequence S a Farey sequence?(1) Sequence S has fewer than 10 elements. Clearly insufficient, S can be any sequence. (2) The second element of sequence S is 1/5 > there is a Farey sequences with 1/5 as the second element in it as well as there are infinitely many other type of sequences also with 1/5 as the second element in it. Not sufficient. (1)+(2) Now, in order 1/5 to be the second element of a Farey sequence, the sequence must be of order 5: {0⁄1, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 1⁄1}, which as we can see has 11 elements, so the sequence which has fewer than 10 elements and has the second element equal to 1/5 cannot be a Farey sequence. Sufficient. Answer: C.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Manager
Joined: 22 Feb 2012
Posts: 85
GMAT 1: 740 Q49 V42 GMAT 2: 670 Q42 V40
GPA: 3.47
WE: Corporate Finance (Aerospace and Defense)

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
29 Mar 2012, 11:23
A Farey sequence of order n is the sequence of fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. For example, the Farey sequence of order 3 is: {0, 1/3 , 1/2, 2/3 , 1}. Is sequence S a Farey sequence?
(1) Sequence S has fewer than 10 elements. (2) The second element of sequence S is 1/5 
I've never seen a question like this... (1) can be any sequence less than 10 elements  insufficient (2) can be any sequence with 2nd term 1/5  insufficient Now combining both statements...The only way to definitively answer is if the Farey sequence (0, 1/5 , .., ..., 1) doesnt fit the parameters given so the Farey sequence is (0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1) Thus the farey sequence in question has 10 elements thus it cannot be the farey sequence so it is sufficient.



Retired Moderator
Status: Enjoying the GMAT journey....
Joined: 26 Aug 2011
Posts: 585
Location: India

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
29 Mar 2012, 18:19
Bunuel wrote: IMPORTANT NOTE: A Farey sequence is not tested on the GMAT, so you can completely ignore this question and skip it.
Thanks Bunnel, I will message the same to Jeff Sackmann also so that he can deleat this question from their 100 challenging questions on Algebra. Regards.
_________________
Fire the final bullet only when you are constantly hitting the Bull's eye, till then KEEP PRACTICING.
A WAY TO INCREASE FROM QUANT 3540 TO 47 : http://gmatclub.com/forum/awaytoincreasefromq3540toq138750.html
Q 47/48 To Q 50 + http://gmatclub.com/forum/thefinalclimbquestforq50fromq47129441.html#p1064367
Three good RC strategies http://gmatclub.com/forum/threedifferentstrategiesforattackingrc127287.html



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8882
Location: Pune, India

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
30 Mar 2012, 04:10
rajeevrks27 wrote: Bunuel wrote: IMPORTANT NOTE: A Farey sequence is not tested on the GMAT, so you can completely ignore this question and skip it.
Thanks Bunnel, I will message the same to Jeff Sackmann also so that he can deleat this question from their 100 challenging questions on Algebra. Regards. GMAT doesn't expect you to know what a Farey sequence is but it could describe the characteristics of the sequence for you and then ask you a question on it as is done in this question. Say, a +1 sequence is a sequence where each term is one of +1 or 1. What is the sum of all the terms of this sequence?... and then two statements or something similar. So I would say that make sure you understand the logic tested in this question.
_________________
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: 52902

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
30 Mar 2012, 04:14
VeritasPrepKarishma wrote: rajeevrks27 wrote: Bunuel wrote: IMPORTANT NOTE: A Farey sequence is not tested on the GMAT, so you can completely ignore this question and skip it.
Thanks Bunnel, I will message the same to Jeff Sackmann also so that he can deleat this question from their 100 challenging questions on Algebra. Regards. GMAT doesn't expect you to know what a Farey sequence is but it could describe the characteristics of the sequence for you and then ask you a question on it as is done in this question. Say, a +1 sequence is a sequence where each term is one of +1 or 1. What is the sum of all the terms of this sequence?... and then two statements or something similar. So I would say that make sure you understand the logic tested in this question. Very true. Though I very much doubt that the GMAT will throw you a question on such a topic as a Farey sequence, which is a bit out of the GMAT scope.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Retired Moderator
Status: Enjoying the GMAT journey....
Joined: 26 Aug 2011
Posts: 585
Location: India

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
30 Mar 2012, 08:41
Thank you Bunnel and Karishma...Great insight by you both...indeed valuable.
_________________
Fire the final bullet only when you are constantly hitting the Bull's eye, till then KEEP PRACTICING.
A WAY TO INCREASE FROM QUANT 3540 TO 47 : http://gmatclub.com/forum/awaytoincreasefromq3540toq138750.html
Q 47/48 To Q 50 + http://gmatclub.com/forum/thefinalclimbquestforq50fromq47129441.html#p1064367
Three good RC strategies http://gmatclub.com/forum/threedifferentstrategiesforattackingrc127287.html



Director
Joined: 27 May 2012
Posts: 681

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
Updated on: 02 Jan 2014, 07:23
Bunuel wrote: From 100 hardest questions Bumping for review and further discussion. Hi Can anyone tell me how the individual terms of a farey sequence are arrived at? Do we have to memorize the terms or is there an actual way to get the terms. I know this may be out of scope for GMAT, but there is no harm in understanding this. Thank you
_________________
 Stne
Originally posted by stne on 27 Dec 2013, 18:50.
Last edited by stne on 02 Jan 2014, 07:23, edited 1 time in total.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8882
Location: Pune, India

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
01 Jan 2014, 22:51
stne wrote: Bunuel wrote: From 100 hardest questions Bumping for review and further discussion. Hi Can anyone tell me how the individual terms of a farey sequence are arrived at? Do we have to memorize the terms or is there an actual way to get the terms. I know this may be out of scope for GMAT, but there is no harm is understanding this. Thank you They have given you the rules: Terms must be between 0 and 1 For order n, the denominator must be less than or equal to n. The terms should be arranged in increasing order. So say we need the Farey sequence of order 2. Terms will be: {0, 1/2, 1} For order 4, terms will be: 0, 1/2, 1, 1/3, 2/3, 1/4, 3/4 Arrange them in increasing order: {0, 1/4, 1/3, 1/2, 2/3, 3/4, 1} Basically, 0 and 1 will always be there (the example shows us that). Then find the terms you will have with each denominator (2, 3, 4 in this case). The terms where the numerator will be less than the denominator will be included. The fractions need to be in the lowest terms. If a fraction is repeated, we put it in only once (we obtained 2/4 above but omitted it since we already have 1/2) With denominator 2, we have only 1/2 With denominator 3, we have 1/3 and 2/3 With denominator 4, we have 1/4, 2/4 (already there), 3/4 Similarly, you can find the sequence of order 5. Note that there will be a unique sequence for every order. Also, you don't need to memorize it since if GMAT does have a question related to any specific sequence, the question will explicitly define the sequence and explain how to arrive at the terms (as done here).
_________________
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 >



Director
Joined: 27 May 2012
Posts: 681

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
02 Jan 2014, 07:15
VeritasPrepKarishma wrote: stne wrote: Bunuel wrote: From 100 hardest questions Bumping for review and further discussion. Hi Can anyone tell me how the individual terms of a farey sequence are arrived at? Do we have to memorize the terms or is there an actual way to get the terms. I know this may be out of scope for GMAT, but there is no harm is understanding this. Thank you They have given you the rules: Terms must be between 0 and 1 For order n, the denominator must be less than or equal to n. The terms should be arranged in increasing order. So say we need the Farey sequence of order 2. Terms will be: {0, 1/2, 1} For order 4, terms will be: 0, 1/2, 1, 1/3, 2/3, 1/4, 3/4 Arrange them in increasing order: {0, 1/4, 1/3, 1/2, 2/3, 3/4, 1} Basically, 0 and 1 will always be there (the example shows us that). Then find the terms you will have with each denominator (2, 3, 4 in this case). The terms where the numerator will be less than the denominator will be included. The fractions need to be in the lowest terms. If a fraction is repeated, we put it in only once (we obtained 2/4 above but omitted it since we already have 1/2) With denominator 2, we have only 1/2 With denominator 3, we have 1/3 and 2/3 With denominator 4, we have 1/4, 2/4 (already there), 3/4 Similarly, you can find the sequence of order 5. Note that there will be a unique sequence for every order. Also, you don't need to memorize it since if GMAT does have a question related to any specific sequence, the question will explicitly define the sequence and explain how to arrive at the terms (as done here). Thank you, here is proof that I have understood it Are the terms of the Farey sequence of order 6 as below,( not arranged in increasing order ). { 0,1/6,1/5,1/4,1/3,1/2,2/3,3/4,2/5,3/5,4/5,5/6,1} Well if at least the terms are correct then I think I have got it, thanks to you but is there an easier way to arrange them in increasing order? All I know of is cross multiplying the fractions and then comparing both , e.g between 2/3 and 3/4 , I cross multiply( denominator * numerator) and get 8 and 9 so I know 3/4 >2/3 , but to use this method with 13 terms seems tedious. So: a) Have I at least got the terms of the farey sequence of order 6 correct? b) Please share any miraculous way you know to arrange them in increasing order ? Thank you +1
_________________
 Stne



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8882
Location: Pune, India

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
02 Jan 2014, 21:22
stne wrote: Thank you, here is proof that I have understood it
Are the terms of the Farey sequence of order 6 as below,( not arranged in increasing order ).
{ 0,1/6,1/5,1/4,1/3,1/2,2/3,3/4,2/5,3/5,4/5,5/6,1}
Well if at least the terms are correct then I think I have got it, thanks to you
but is there an easier way to arrange them in increasing order? All I know of is cross multiplying the fractions and then comparing both , e.g between 2/3 and 3/4 , I cross multiply( denominator * numerator) and get 8 and 9 so I know 3/4 >2/3 , but to use this method with 13 terms seems tedious. So: a) Have I at least got the terms of the farey sequence of order 6 correct? b) Please share any miraculous way you know to arrange them in increasing order ?
Thank you +1 Yes, the terms are correct. I don't really have a miraculous method to arrange them though some logic goes a long way in reducing our work. Two things help us here: 1. Many fractions already have the same numerator or denominator. We can compare them by just looking at them. Same numerator  Larger the denominator, smaller the fraction. Same denominator  Larger the numerator, larger the fraction. 2. The fractions are simple so we already know their percent equivalents. 2/3 is 66.7%; 3/5 = 60%, 3/4 is 75%, 5/6 is 83%; 2/5 is 40%; 1/3 is 33% etc First split them into two groups  less than 1/2 and greater than 1/2 Now just arrange them according to the discussion above. { 0,1/6,1/5,1/4,1/3, 2/5,1/2,3/5,2/3,3/4,4/5,5/6,1}
_________________
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 >



Director
Joined: 27 May 2012
Posts: 681

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
03 Jan 2014, 07:52
VeritasPrepKarishma wrote: stne wrote: Thank you, here is proof that I have understood it
Are the terms of the Farey sequence of order 6 as below,( not arranged in increasing order ).
{ 0,1/6,1/5,1/4,1/3,1/2,2/3,3/4,2/5,3/5,4/5,5/6,1}
Well if at least the terms are correct then I think I have got it, thanks to you
but is there an easier way to arrange them in increasing order? All I know of is cross multiplying the fractions and then comparing both , e.g between 2/3 and 3/4 , I cross multiply( denominator * numerator) and get 8 and 9 so I know 3/4 >2/3 , but to use this method with 13 terms seems tedious. So: a) Have I at least got the terms of the farey sequence of order 6 correct? b) Please share any miraculous way you know to arrange them in increasing order ?
Thank you +1 Yes, the terms are correct. I don't really have a miraculous method to arrange them though some logic goes a long way in reducing our work. Two things help us here: 1. Many fractions already have the same numerator or denominator. We can compare them by just looking at them. Same numerator  Larger the denominator, smaller the fraction. Same denominator  Larger the numerator, larger the fraction. 2. The fractions are simple so we already know their percent equivalents. 2/3 is 66.7%; 3/5 = 60%, 3/4 is 75%, 5/6 is 83%; 2/5 is 40%; 1/3 is 33% etc First split them into two groups  less than 1/2 and greater than 1/2 Now just arrange them according to the discussion above. { 0,1/6,1/5,1/4,1/3, 2/5,1/2,3/5,2/3,3/4,4/5,5/6,1} Thank you, I think you may have unknowingly lit a bulb in me. the idea of converting fractions to percents in order to compare them never dawned on me. I always used to cross multiply or convert to common denominator to compare, now I can convert to percent too! I think you have given the miraculous way I was searching for. also the same denominator and same numerator hint has really helped. below is proof of my understanding,it is only because of percent conversion as shown by you, I could attempt something as herculean as this , earlier I couldn't even have imagined attempting something as tedious as this.Thank you. Farey sequence of order 7 in increasing order. {0, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1} Hope these are correct, full credit to you, thank you a ton, +1
_________________
 Stne



Intern
Joined: 24 Jun 2018
Posts: 31

A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
07 Oct 2018, 06:30
Bunuel wrote: A Farey sequence of order n is the sequence of fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. For example, the Farey sequence of order 3 is: {0, 1/3 , 1/2, 2/3 , 1}. Is sequence S a Farey sequence?
(1) Sequence S has fewer than 10 elements. Clearly insufficient, S can be any sequence. (2) The second element of sequence S is 1/5 > there is a Farey sequences with 1/5 as the second element in it as well as there are infinitely many other type of sequences also with 1/5 as the second element in it. Not sufficient.
(1)+(2) Now, in order 1/5 to be the second element of a Farey sequence, the sequence must be of order 5: {0⁄1, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 1⁄1}, which as we can see has 11 elements, so the sequence which has fewer than 10 elements and has the second element equal to 1/5 cannot be a Farey sequence. Sufficient.
Answer: C. (2) The second element of sequence S is 1/5 > there is a Farey sequence with 1/5 as the second element in it as well as there are infinitely many other type of sequences also with 1/5 as the second element in it. Not sufficient.. I think there can only be one such sequence in which 1/5 is the second number coz the first digit will be 0 and if 1/5 is the second number then it is sure that S is not greater than 5. Please clarify, I didn't understand this.
_________________
A sentence which can make a happy person sad and a sad person happy 'this TIME will change'



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 730

A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
07 Oct 2018, 12:40
rajeevrks27 wrote: A Farey sequence of order n is the sequence of fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. For example, the Farey sequence of order 3 is: {0, 1/3 , 1/2, 2/3 , 1}. Is sequence S a Farey sequence?
(1) Sequence S has fewer than 10 elements. (2) The second element of sequence S is 1/5
Important: 1. "Fractions between 0 and 1" must (and will) be considered as "fractions between 0 and 1, both included" so that the example given satisfies the definition presented. 2. The { } notation will always denote here a finite sequence. (In other words, the order of the elements presented is relevant.) \(S\,\,\mathop = \limits^? \,\,{\rm{Farey}}\) \(\left( 1 \right)\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,S = \left\{ {0;{1 \over 3};{1 \over 2};{2 \over 3};1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {{\rm{given}}} \right)\,\, \hfill \cr \,{\rm{Take}}\,\,S = \left\{ {0;1;{1 \over 3};{1 \over 2};{2 \over 3}} \right\}\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\left( {{\rm{wrong}}\,\,{\rm{order}}} \right)\, \hfill \cr} \right.\) \(\left( 2 \right)\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,S = \left\{ {0;{1 \over 5};{2 \over 5};{3 \over 5};{4 \over 5};1;{1 \over 4};{2 \over 4} = {1 \over 2};{3 \over 4};{1 \over 3};{2 \over 3}} \right\}\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\left( {{\rm{wrong}}\,\,{\rm{order, }}\,{\rm{although}}\,\,{1 \over 5}\,\,{\rm{IS}}\,\,{\rm{the}}\,{\rm{ second}}\,\,{\rm{here!}}} \right)\,\, \hfill \cr \,{\rm{Take}}\,\,S = \left\{ {0;{1 \over 5};{2 \over 5};{3 \over 5};{4 \over 5};1;{1 \over 4};{2 \over 4} = {1 \over 2};{3 \over 4};{1 \over 3};{2 \over 3}} \right\}\,\,{\rm{but}}\,\,{\rm{rewritten}}\,\,\,{\rm{in}}\,\,{\rm{increasing}}\,\,\,{\rm{order}}\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\left( {{1 \over 5}\,\,{\rm{WILL}}\,\,{\rm{BE}}\,\,{\rm{the}}\,\,{\rm{second!}}} \right)\,\, \hfill \cr} \right.\) \(\left( {1 + 2} \right)\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\left( {{1 \over 5} \in \,\,S\,\,{\rm{Farey}}\,\,\,\, \Rightarrow \,\,\,{\rm{S}}\,\,{\rm{has}}\,\,{\rm{more}}\,\,{\rm{than}}\,\,{\rm{10}}\,\,{\rm{elements}}!} \right)\) This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net



DS Forum Moderator
Joined: 21 Aug 2013
Posts: 1431
Location: India

Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
Show Tags
07 Oct 2018, 20:42
ReadyPlayerOne wrote: Bunuel wrote: A Farey sequence of order n is the sequence of fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. For example, the Farey sequence of order 3 is: {0, 1/3 , 1/2, 2/3 , 1}. Is sequence S a Farey sequence?
(1) Sequence S has fewer than 10 elements. Clearly insufficient, S can be any sequence. (2) The second element of sequence S is 1/5 > there is a Farey sequences with 1/5 as the second element in it as well as there are infinitely many other type of sequences also with 1/5 as the second element in it. Not sufficient.
(1)+(2) Now, in order 1/5 to be the second element of a Farey sequence, the sequence must be of order 5: {0⁄1, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 1⁄1}, which as we can see has 11 elements, so the sequence which has fewer than 10 elements and has the second element equal to 1/5 cannot be a Farey sequence. Sufficient.
Answer: C. (2) The second element of sequence S is 1/5 > there is a Farey sequence with 1/5 as the second element in it as well as there are infinitely many other type of sequences also with 1/5 as the second element in it. Not sufficient.. I think there can only be one such sequence in which 1/5 is the second number coz the first digit will be 0 and if 1/5 is the second number then it is sure that S is not greater than 5. Please clarify, I didn't understand this. Hello In the second statement, we are given that 1/5 is the second term of sequence S. But we cannot already assume that S is a Farey sequence. There can be many sequences in this world where 1/5 is the second term of that sequence without it being a Farey sequence. Eg, there can be an AP.. (1/10, 1/5, 3/10, 2/5... this has a common difference of 1/10) or there can be a GP.. (1, 1/5, 1/25, 1/125,... this has a common ratio of 1/5) Thats why there can be infinite sequences with second term as 1/5, we cannot say on the basis of that whether S is a Farey sequence or not.




Re: A Farey sequence of order n is the sequence of fractions
[#permalink]
07 Oct 2018, 20:42






