GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 15 Oct 2019, 15:25

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Ax(y) is an operation that adds 1 to y

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Manager
Manager
avatar
Joined: 29 Nov 2011
Posts: 71
Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 03 Jul 2012, 18:17
1
23
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

35% (02:47) correct 65% (02:36) wrong based on 304 sessions

HideShow timer Statistics

Ax(y) is an operation that adds 1 to y and then multiplies the result by x. If x = 2/3, then Ax(Ax(Ax(Ax(Ax(x))))) is between

(A) 0 and ½
(B) ½ and 1
(C) 1 and 1½
(D) 1½ and 2
(E) 2 and 2½
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58340
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 04 Jul 2012, 02:32
16
5
Smita04 wrote:
Ax(y) is an operation that adds 1 to y and then multiplies the result by x. If x = 2/3, then Ax(Ax(Ax(Ax(Ax(x))))) is between

(A) 0 and ½
(B) ½ and 1
(C) 1 and 1½
(D) 1½ and 2
(E) 2 and 2½

Sorry, but I don't have the OA.


The question should read:

\(A_x(y)\) is an operation that adds 1 to y and then multiplies the result by x. If x = 2/3, then \(A_x(A_x(A_x(A_x(A_x(x)))))\) is between

(A) 0 and ½
(B) ½ and 1
(C) 1 and 1½
(D) 1½ and 2
(E) 2 and 2½

According to the stem:
\(A_x(x)=(x+1)x=x^2+x\);
\(A_x(A_x(x))=A_x(x^2+x)=(x^2+x+1)*x=x^3+x^2+x\);
\(A_x(A_x(A_x(x)))=A_x(x^3+x^2+x)=(x^3+x^2+x+1)*x=x^4+x^3+x^2+x\);
...

We can see the pattern now, so \(A_x(A_x(A_x(A_x(A_x(x)))))=x^6+x^5+x^4+x^3+x^2+x\).

For \(x=\frac{2}{3}\) we'll get: \((\frac{2}{3})^6+(\frac{2}{3})^5+(\frac{2}{3})^4+(\frac{2}{3})^3+(\frac{2}{3})^2+(\frac{2}{3})\).

So, we have the sum of the 6 terms of the geometric progression with the first term equal to \(\frac{2}{3}\) and the common ratio also equal to \(\frac{2}{3}\).

Now, the sum of infinite geometric progression with common ratio \(|r|<1\), is \(sum=\frac{b}{1-r}\), where \(b\) is the first term. So, if we had infinite geometric progression instead of just 6 terms then its sum would be \(Sum=\frac{\frac{2}{3}}{1-\frac{2}{3}}=2\). Which means that the sum of this sequence will never exceed 2, also as we have big enough number of terms (6) then the sum will be very close to 2, so we can safely choose answer choice D.

Answer: D.

One can also use direct formula.
We have geometric progression with \(b=\frac{2}{3}\), \(r=\frac{2}{3}\) and \(n=6\);

\(S_n=\frac{b(1-r^n)}{(1-r)}\) --> \(S_{6}=\frac{\frac{2}{3}(1-(\frac{2}{3})^{6})}{(1-\frac{2}{3})}=2*(1-(\frac{2}{3})^{6})\). Since \((\frac{2}{3})^{6}\) is very small number then \(1-(\frac{2}{3})^{6}\) will be less than 1 but very close to it, hence \(2*(1-(\frac{2}{3})^{6})\) will be less than 2 but very close to it.

Answer: D.

Hope it's clear.
_________________
General Discussion
Intern
Intern
avatar
Joined: 14 Apr 2012
Posts: 10
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 03 Jul 2012, 21:51
If I read the question correctly, AX(Y) = (1+Y)X
Putting X=2/3,
=> AX(X) = (1+ 2/3)*(2/3)
=> AX(X) = (5/3)*(2/3)
=> AX(X) = 10/9 > 1

AX(AX(X) = (1+10/9)(10/9)....The first term in the bracket will be greater than 2 and next term is greater than one...so the product is greater than 2.
The next operation will make the (1+Y) term greater than 3, so basically as per my understanding there is some information missing in the question or I misunderstood.

Regards,
Tushar
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58340
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 21 Jun 2013, 03:47
Intern
Intern
avatar
Joined: 22 May 2013
Posts: 43
Concentration: General Management, Technology
GPA: 3.9
WE: Information Technology (Computer Software)
GMAT ToolKit User
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 22 Jun 2013, 02:27
Smita04 wrote:
Ax(y) is an operation that adds 1 to y and then multiplies the result by x. If x = 2/3, then Ax(Ax(Ax(Ax(Ax(x))))) is between

(A) 0 and ½
(B) ½ and 1
(C) 1 and 1½
(D) 1½ and 2
(E) 2 and 2½


This one was quite tricky.. lovely explanation by Bunuel using geometric progression.
_________________
PS: Like my approach? Please Help me with some Kudos. :-)
Veritas Prep GMAT Instructor
User avatar
V
Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 15 Jan 2014, 21:56
1
tusharacc wrote:
If I read the question correctly, AX(Y) = (1+Y)X
Putting X=2/3,
=> AX(X) = (1+ 2/3)*(2/3)
=> AX(X) = (5/3)*(2/3)
=> AX(X) = 10/9 > 1

AX(AX(X) = (1+10/9)(10/9)....The first term in the bracket will be greater than 2 and next term is greater than one...so the product is greater than 2.
The next operation will make the (1+Y) term greater than 3, so basically as per my understanding there is some information missing in the question or I misunderstood.

Regards,
Tushar


The highlighted portion is incorrect here.

Ax(Ax(X)) = (1+10/9)(2/3)
Note that X is 2/3 only. Ax(X) is 10/9.
Ax(10/9) = (1 + 10/9)(2/3)
_________________
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 >
Intern
Intern
avatar
Joined: 22 Apr 2015
Posts: 11
GMAT 1: 760 Q50 V44
Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 15 May 2015, 11:04
Bunuel

I solved this in about 1:20 by considering that eventually, this progression would stabilize, and so for any number Ax(Ax(Ax(...(Ax(y))) always approaches a number.

So I set up the equation:
y = (y+1) * (2/3)
3/2y = y + 1
1/2y = 1
y = 2

So the equation will always approach the number 2.

Since we can see that the equation is approaching 2 from below, it must be close to 2, but just slightly less, hence D
Intern
Intern
avatar
B
Joined: 06 Apr 2017
Posts: 29
Location: United States (OR)
Concentration: Finance, Leadership
Schools: Haas EWMBA '21
GMAT 1: 730 Q48 V44
GMAT 2: 730 Q49 V40
GPA: 3.98
WE: Corporate Finance (Health Care)
GMAT ToolKit User
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 27 Apr 2017, 16:25
Bunuel, you never cease to amaze me with the depth of your abstract and theoretical solutions.

The simplicity of the numbers in this question and the depth of recursion in the function make it possible to calculate the value directly. This approach will not always be viable with similar questions.

\(Ax(x) = \frac{2}{3} * (\frac{2}{3} + 1) = \frac{10}{9}\)
\(Ax(10/9) = \frac{2}{3} * (\frac{10}{9} +1) = \frac{38}{27}\)
\(Ax(38/27) = \frac{2}{3} * (\frac{38}{27} + 1) = \frac{130}{81}\)
\(Ax(130/81) = \frac{2}{3} * (\frac{130}{81} +1) = \frac{422}{243}\)
\(Ax(422/243) = \frac{2}{3} * (\frac{422}{243} + 1) = \frac{1330}{729}\)
\(1.5 < \frac{1330}{729} < 2\)

Answer d
Senior Manager
Senior Manager
avatar
G
Status: Countdown Begins...
Joined: 03 Jul 2016
Posts: 277
Location: India
Concentration: Technology, Strategy
Schools: IIMB
GMAT 1: 580 Q48 V22
GPA: 3.7
WE: Information Technology (Consulting)
GMAT ToolKit User Reviews Badge
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 28 Apr 2017, 08:48
spence11 wrote:
Bunuel, you never cease to amaze me with the depth of your abstract and theoretical solutions.

The simplicity of the numbers in this question and the depth of recursion in the function make it possible to calculate the value directly. This approach will not always be viable with similar questions.

\(Ax(x) = \frac{2}{3} * (\frac{2}{3} + 1) = \frac{10}{9}\)
\(Ax(10/9) = \frac{2}{3} * (\frac{10}{9} +1) = \frac{38}{27}\)
\(Ax(38/27) = \frac{2}{3} * (\frac{38}{27} + 1) = \frac{130}{81}\)
\(Ax(130/81) = \frac{2}{3} * (\frac{130}{81} +1) = \frac{422}{243}\)
\(Ax(422/243) = \frac{2}{3} * (\frac{422}{243} + 1) = \frac{1330}{729}\)
\(1.5 < \frac{1330}{729} < 2\)

Answer d


You should multiply with the updated number every time and not \(\frac{2}{3}\)
Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 13167
Re: Ax(y) is an operation that adds 1 to y  [#permalink]

Show Tags

New post 11 Oct 2019, 22:58
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.
_________________
GMAT Club Bot
Re: Ax(y) is an operation that adds 1 to y   [#permalink] 11 Oct 2019, 22:58
Display posts from previous: Sort by

Ax(y) is an operation that adds 1 to y

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne