January 17, 2019 January 17, 2019 08:00 AM PST 09:00 AM PST Learn the winning strategy for a high GRE score — what do people who reach a high score do differently? We're going to share insights, tips and strategies from data we've collected from over 50,000 students who used examPAL. January 19, 2019 January 19, 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

Senior Manager
Status: Up again.
Joined: 31 Oct 2010
Posts: 492
Concentration: Strategy, Operations
GMAT 1: 710 Q48 V40 GMAT 2: 740 Q49 V42

Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
02 Jan 2011, 01:34
Question Stats:
65% (01:45) correct 35% (01:41) wrong based on 339 sessions
HideShow timer Statistics
Is \(1+x+x^2+x^3+....+x^{10}\)positive? (1) \(x<1\) (2) \(x^2>2\) Source: GMAT Club Hardest problems.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
My GMAT debrief: http://gmatclub.com/forum/from620to710mygmatjourney114437.html




Math Expert
Joined: 02 Sep 2009
Posts: 52187

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
02 Jan 2011, 02:14
gmatpapa wrote: Is \(1+x+x^2+x^3+....+x^{10}\)positive?
1. \(x<1\) 2. \(x^2>2\)
Source: GMAT Club Hardest problems. Is \(1+(x+x^2)+(x^3+x^4)+...+(x^9+x^{10})>0\)? (1) \(x<1\): \(x+x^2>0\) (x<1 meas that x^2>x), \(x^3+x^4>0\), ..., \(x^9+x^{10}>0\), so the sum is also more than zero. Sufficient. (2) \(x^2>2\): even if x itself is negative then still as above: \(x+x^2>0\), \(x^3+x^4>0\), ..., \(x^9+x^{10}>0\), so the sum is also more than zero. Sufficient. Answer: D.
_________________
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




Senior Manager
Status: Up again.
Joined: 31 Oct 2010
Posts: 492
Concentration: Strategy, Operations
GMAT 1: 710 Q48 V40 GMAT 2: 740 Q49 V42

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
03 Jan 2011, 10:34
Its clear now! Thanks!!
_________________
My GMAT debrief: http://gmatclub.com/forum/from620to710mygmatjourney114437.html



Intern
Joined: 20 Jul 2010
Posts: 3

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
04 Jan 2011, 05:56
Hi! This is a geometric progression. Could you explain this using the sum formula?
Thanks!



Math Expert
Joined: 02 Sep 2009
Posts: 52187

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
04 Jan 2011, 06:36
amankalra wrote: Hi! This is a geometric progression. Could you explain this using the sum formula?
Thanks! Sum of the first \(n\) terms of geometric progression is given by: \(sum=\frac{b*(r^{n}1)}{r1}\), where \(b\) is the first term, \(n\) # of terms and \(r\) is a common ratio \(\neq{1}\). So, in our case \(b=x\) and \(r=x\) > \(1+(x+x^2+x^3+x^4+...+x^9+x^{10})=1+sum_{10}=1+\frac{x*(x^{10}1)}{x1}\). (1) \(x<1\) > \(1+\frac{x*(x^{10}1)}{x1}=1+\frac{negative*positive}{negative}=1+positive=positive\). Sufficient. (2) \(x^2>2\) > \(x<\sqrt{2}\) or \(x>\sqrt{2}\) > so, either \(1+\frac{x*(x^{10}1)}{x1}=1+\frac{negative*positive}{negative}=1+positive=positive\) or \(1+\frac{x*(x^{10}1)}{x1}=1+\frac{positive*positive}{positive}=1+positive=positive\). Sufficient. Answer: D. Hope it's clear.
_________________
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



Intern
Joined: 20 Jul 2010
Posts: 3

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
04 Jan 2011, 06:54
Thanks! I was actually considering a total of 11 terms, and b=1. Is that right?



Math Expert
Joined: 02 Sep 2009
Posts: 52187

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
04 Jan 2011, 07:04



Intern
Joined: 20 Jul 2010
Posts: 3

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
04 Jan 2011, 07:08
Okay. Thanks a ton!



Manager
Joined: 17 Dec 2010
Posts: 91
Location: Australia
GPA: 3.37
WE: Engineering (Consulting)

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
04 Jan 2011, 20:14
I went for the 11 second solution and picked A... If I took time on it and actualyl solved it  it is clear that the answer is D. I got tripped up be thinking to myself "x could have two values > not sufficient" solution would have been let x = 2 or 2. 2 or 2^10 = 1024. 11 terms in series. Avg value of terms is 1025 => sum is 1025 / 2 = positive > Sufficient WHY OH WHY DO I RUSH?!?!?!
_________________
Kudos always appreciated if my post helped you



Intern
Joined: 18 Mar 2012
Posts: 47
GPA: 3.7

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
25 Jan 2013, 05:05
Is 1 + x + x^2 + … + x^10 positive?
1) x < 1 2) x^2 > 2



Math Expert
Joined: 02 Sep 2009
Posts: 52187

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
25 Jan 2013, 05:34



Manager
Joined: 02 Jan 2013
Posts: 57
GPA: 3.2
WE: Consulting (Consulting)

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
26 Jan 2013, 04:40
This is another one of those weird problems, where neither (1) nor (2) are necessary.
The expression 1 + x + xˆ2+...+ x^10 is ALWAYS positive, for any real value of x. PERIOD.
The best way to prove this, is transforming the expression in the GP sum (xˆ11  1)/(x1) already mentioned.



Manager
Joined: 10 Mar 2014
Posts: 191

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
27 Apr 2014, 09:59
Bunuel wrote: gmatpapa wrote: Is \(1+x+x^2+x^3+....+x^{10}\)positive?
1. \(x<1\) 2. \(x^2>2\)
Source: GMAT Club Hardest problems. Is \(1+(x+x^2)+(x^3+x^4)+...+(x^9+x^{10})>0\)? (1) \(x<1\): \(x+x^2>0\) (x<1 meas that x^2>x), \(x^3+x^4>0\), ..., \(x^9+x^{10}>0\), so the sum is also more than zero. Sufficient. (2) \(x^2>2\): even if x itself is negative then still as above: \(x+x^2>0\), \(x^3+x^4>0\), ..., \(x^9+x^{10}>0\), so the sum is also more than zero. Sufficient. Answer: D. Hi Bunnel, According to st1 (x<1) so can write this as 1+ (2) + (2)^2+ (2)^3+ (2)^10 can I use above as geometric progression. ( dont include 1 in seried we will add it lastly) If I will use this in gp then I will get result as <0 bcz first term is ve Please clarify. Thanks



Director
Affiliations: GMATQuantum
Joined: 19 Apr 2009
Posts: 612

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
27 Apr 2014, 12:24
I agree with @caioguima, this is not a very good problem. GMAT won't give you a problem where the statements are completely redundant, and the answer to the question is already a definite Yes. At least, I haven't seen an official GMAT question that follows this format, please correct me if anyone has seen such an example.
To expand on @caioguima, the expression 1+x+x^2+x^3+....+x^10 is positive for all values of x greater than or equal to zero. If we rewrite this expression using the geometric series format, it becomes (x^111)/(x1)[skipping those details here], and if we now consider the case of x<0, then both numerator and denominator are negative, making the expression positive for all values of x<0. Therefore, 1+x+x^2+x^3+....+x^10 is positive for all values of x. And the statements become redundant at this stage, which I have never seen on the GMAT.
Cheers, Dabral



NonHuman User
Joined: 09 Sep 2013
Posts: 9420

Re: Is 1+x+x^2+x^3+....+x^10 positive?
[#permalink]
Show Tags
13 Mar 2018, 04:15
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




Re: Is 1+x+x^2+x^3+....+x^10 positive? &nbs
[#permalink]
13 Mar 2018, 04:15






