Find all School-related info fast with the new School-Specific MBA Forum

It is currently 23 Oct 2014, 10:09

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10)

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Manager
Manager
avatar
Joined: 20 Oct 2011
Posts: 123
Location: Canada
Concentration: Sustainability, General Management
GMAT 1: 710 Q49 V38
GPA: 3.98
Followers: 1

Kudos [?]: 13 [0], given: 11

3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10) [#permalink] New post 15 Nov 2011, 07:22
1
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

54% (03:23) correct 46% (02:12) wrong based on 96 sessions
\frac{3(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = ?

(A) 2^{10} + 2^5
(B) 2^{10} + 2^6
(C) 2^{11} + 2^5
(D) 2^{11} + 2^6
(E) 2^{16}


I was stuck at this question for a loooong time and I had to check the solution to see how it was done. Got it from the GMAT Quantum site. I haven't come across solving for sum of GP like this before in any of the Manhattan guides or any other GMAT question. Damn!
[Reveal] Spoiler: OA

Last edited by alinomoto on 15 Nov 2011, 10:49, edited 2 times in total.
Kaplan Promo CodeKnewton GMAT Discount CodesVeritas Prep GMAT Discount Codes
2 KUDOS received
Manager
Manager
avatar
Joined: 29 Oct 2011
Posts: 188
Concentration: General Management, Technology
Schools: Sloan '16 (D)
GMAT 1: 760 Q49 V44
GPA: 3.76
Followers: 7

Kudos [?]: 73 [2] , given: 19

Re: Difficult Geometric series question. [#permalink] New post 15 Nov 2011, 09:16
2
This post received
KUDOS
I used mostly brute force, with some factoring, to solve this. I'd be interested in a more elegant solution.

\frac{3*(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = \frac{3*4^5(1+4^1+4^2+4^3+4^4+4^5)}{2^5*(1+2^1+2^2+2^3+2^4+2^5)} = \frac{3*2^5*2^5(1+4+16+64+256+1024)}{2^5*(1+2+4+8+16+32)} =

=\frac{3*2^5*1365}{63} = \frac{2^5*1365}{21} = 2^5*65 = 2^5*(64+1) = 2^5*(2^6+1) = 2^{11}+2^5
3 KUDOS received
Manager
Manager
avatar
Joined: 20 Oct 2011
Posts: 123
Location: Canada
Concentration: Sustainability, General Management
GMAT 1: 710 Q49 V38
GPA: 3.98
Followers: 1

Kudos [?]: 13 [3] , given: 11

Re: Difficult Geometric series question. [#permalink] New post 15 Nov 2011, 10:30
3
This post received
KUDOS
kostyan5 wrote:
I used mostly brute force, with some factoring, to solve this. I'd be interested in a more elegant solution.

\frac{3*(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = \frac{3*4^5(1+4^1+4^2+4^3+4^4+4^5)}{2^5*(1+2^1+2^2+2^3+2^4+2^5)} = \frac{3*2^5*2^5(1+4+16+64+256+1024)}{2^5*(1+2+4+8+16+32)} =

=\frac{3*2^5*1365}{63} = \frac{2^5*1365}{21} = 2^5*65 = 2^5*(64+1) = 2^5*(2^6+1) = 2^{11}+2^5


Yes, that is the way I was going at it too, but I realised that there has to be an easier way to do this under 2 minutes.

\frac{3*(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})}

just consider the top and bottom halves of the geometric progression.

S1 = (4^5+4^6+4^7+4^8+4^9+4^{10})

and S2 = (2^5+2^6+2^7+2^8+2^9+2^{10})

Solving for S1, since the common fraction in the GP is 4, multiply S1 by 4.

Thus, 4S1 = 4*(4^5+4^6+4^7+4^8+4^9+4^{10}) = (4^6+4^7+4^8+4^9+4^{10}+4^{11})

Subtract 4S1 - S1 = 3S1 = (4^{11} - 4^5)

or S1 = \frac{(4^{11} - 4^5)}{3}

Similarly, for S2, multiply by common factor (2) and subtract:

2S2 - S2 = (2^{11} - 2^5)

Notice that 3S1 is the numerator cancels with the 3 of S1.

Thus we get \frac{3 *(4^{11} - 4^5)}{3} * \frac{1}{(2^{11} - 2^5)}.

Solving and factoring:
\frac{(4^{11} - 4^5)}{(2^{11} - 2^5)} = \frac{((2^2)^{11} - (2^2)^5)}{(2^{11} - 2^5)} = \frac{((2^{11})^2 - (2^5)^2)}{(2^{11} - 2^5)}

Now you see that the numerator takes the form a^2 - b^2 = (a+b) (a-b)

Solve and you get answer (C) 2^{11} + 2^5
Manager
Manager
avatar
Joined: 29 Oct 2011
Posts: 188
Concentration: General Management, Technology
Schools: Sloan '16 (D)
GMAT 1: 760 Q49 V44
GPA: 3.76
Followers: 7

Kudos [?]: 73 [0], given: 19

Re: Difficult Geometric series question. [#permalink] New post 15 Nov 2011, 13:53
Thanks. Neat trick.
Expert Post
6 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4877
Location: Pune, India
Followers: 1154

Kudos [?]: 5369 [6] , given: 165

Re: Difficult Geometric series question. [#permalink] New post 15 Nov 2011, 20:31
6
This post received
KUDOS
Expert's post
1
This post was
BOOKMARKED
alinomoto wrote:
\frac{3(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = ?

(A) 2^{10} + 2^5
(B) 2^{10} + 2^6
(C) 2^{11} + 2^5
(D) 2^{11} + 2^6
(E) 2^{16}


I was stuck at this question for a loooong time and I had to check the solution to see how it was done. Got it from the GMAT Quantum site. I haven't come across solving for sum of GP like this before in any of the Manhattan guides or any other GMAT question. Damn!


It is certainly not a GMAT-type question so I wouldn't worry about it.
Though, you can do it quickly using the formula Sum of GP = a(r^n - 1)/(r - 1)
a = first term, r = common ratio (which is 4 in the numerator) and n = number of terms

(4^5+4^6+4^7+4^8+4^9+4^{10}) = 4^5(4^6 - 1)/(4-1) = 4^5(4^6 - 1)/3

(2^5+2^6+2^7+2^8+2^9+2^{10}) = 2^5(2^6 - 1)/(2-1)

The required fraction becomes: \frac{4^5(4^6 - 1)}{2^5(2^6 - 1)}

= \frac{2^5(2^6 + 1)(2^6 - 1)}{(2^6 - 1)} (as done above)
= 2^{11} + 2^5
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

CEO
CEO
User avatar
Joined: 09 Sep 2013
Posts: 2834
Followers: 207

Kudos [?]: 42 [0], given: 0

Premium Member
Re: 3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10) [#permalink] New post 16 Sep 2013, 03:50
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 Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Expert Post
1 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 23398
Followers: 3610

Kudos [?]: 28834 [1] , given: 2854

Re: 3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10) [#permalink] New post 16 Sep 2013, 04:00
1
This post received
KUDOS
Expert's post
alinomoto wrote:
\frac{3(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = ?

(A) 2^{10} + 2^5
(B) 2^{10} + 2^6
(C) 2^{11} + 2^5
(D) 2^{11} + 2^6
(E) 2^{16}


I was stuck at this question for a loooong time and I had to check the solution to see how it was done. Got it from the GMAT Quantum site. I haven't come across solving for sum of GP like this before in any of the Manhattan guides or any other GMAT question. Damn!


Similar questions to practice:
3-108690.html
new-tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029222
sequence-s-is-defined-as-follows-s1-2-s2-2-1-s3-2-2-sn-129435.html
2-99058.html
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

VP
VP
User avatar
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1086
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Followers: 6

Kudos [?]: 304 [0], given: 170

Re: 3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10) [#permalink] New post 26 Feb 2014, 20:30
Just want to confirm, is this a GMAT question??
_________________

Kindly press "+1 Kudos" to appreciate :)

Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 23398
Followers: 3610

Kudos [?]: 28834 [0], given: 2854

Re: 3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10) [#permalink] New post 27 Feb 2014, 04:23
Expert's post
PareshGmat wrote:
Just want to confirm, is this a GMAT question??


Please read Karishma's response:
VeritasPrepKarishma wrote:
alinomoto wrote:
\frac{3(4^5+4^6+4^7+4^8+4^9+4^{10})}{(2^5+2^6+2^7+2^8+2^9+2^{10})} = ?

(A) 2^{10} + 2^5
(B) 2^{10} + 2^6
(C) 2^{11} + 2^5
(D) 2^{11} + 2^6
(E) 2^{16}


It is certainly not a GMAT-type question so I wouldn't worry about it.
...

_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

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?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Manager
Manager
User avatar
Joined: 19 Apr 2009
Posts: 178
Location: San Francisco, California
Followers: 50

Kudos [?]: 180 [0], given: 1

Re: 3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10) [#permalink] New post 30 Mar 2014, 08:09
@PareshGMAT

I wrote this question and I want to assure everyone that this concept has been tested on the GMAT. However, one will encounter this question only if scoring close to Q50/Q51. I would recommend understanding the approach that @alinomoto took by multiplying the geometric series by the common factor and subtracting it. This same approach is used to obtain a general expression for the sum of the terms in a geometric sequence. One could memorize the formula for a geometric progression, but I personally think it is better to understand the underlying approach that is used to arrive at the expression.

Over the last four or five years, the GMAT test writers have had to introduce newer concepts to distinguish between students scoring at the upper percentile on the GMAT. The GMAT quant scores have been steadily creeping up over the last decade or so, the fact that Q51 is now 98% as opposed to 99% reflects this. The introduction of summation of a geometric sequence is an example of such a topic.

Cheers,
Dabral
_________________

Free Video Explanations: OFFICIAL GUIDE GMAT 13, 12, 11, 10; QUANT REVIEW 2nd, 1st.

Re: 3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10)   [#permalink] 30 Mar 2014, 08:09
Display posts from previous: Sort by

3(4^5+4^6+4^7+4^8+4^9+4^10)/(2^5+2^6+2^7+2^8+2^9+2^10)

  Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Privacy Policy| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.