Is 1+x+x^2+x^3+....+x^10 positive? : GMAT Data Sufficiency (DS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 22 Jan 2017, 10:43

### 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

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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Is 1+x+x^2+x^3+....+x^10 positive?

Author Message
TAGS:

### Hide Tags

Current Student
Status: Up again.
Joined: 31 Oct 2010
Posts: 541
Concentration: Strategy, Operations
GMAT 1: 710 Q48 V40
GMAT 2: 740 Q49 V42
Followers: 21

Kudos [?]: 411 [1] , given: 75

### Show Tags

02 Jan 2011, 01:34
1
KUDOS
4
This post was
BOOKMARKED
00:00

Difficulty:

35% (medium)

Question Stats:

59% (02:07) correct 41% (01:06) wrong based on 284 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.
[Reveal] Spoiler: OA

_________________

My GMAT debrief: http://gmatclub.com/forum/from-620-to-710-my-gmat-journey-114437.html

Math Expert
Joined: 02 Sep 2009
Posts: 36598
Followers: 7095

Kudos [?]: 93457 [1] , given: 10563

### Show Tags

02 Jan 2011, 02:14
1
KUDOS
Expert's post
7
This post was
BOOKMARKED
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.

_________________
Current Student
Status: Up again.
Joined: 31 Oct 2010
Posts: 541
Concentration: Strategy, Operations
GMAT 1: 710 Q48 V40
GMAT 2: 740 Q49 V42
Followers: 21

Kudos [?]: 411 [0], given: 75

### Show Tags

03 Jan 2011, 10:34
Its clear now! Thanks!!
_________________

My GMAT debrief: http://gmatclub.com/forum/from-620-to-710-my-gmat-journey-114437.html

Intern
Joined: 20 Jul 2010
Posts: 5
Followers: 0

Kudos [?]: 0 [0], given: 7

### 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: 36598
Followers: 7095

Kudos [?]: 93457 [1] , given: 10563

### Show Tags

04 Jan 2011, 06:36
1
KUDOS
Expert's post
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: $$sum=\frac{b*(r^{n}-1)}{r-1}$$, where $$b$$ is the first term, $$n$$ # of terms and $$r$$ is a common ratio $$\neq{1}$$.

So, in our case $$b=x$$ and $$r=x$$ --> $$1+(x+x^2+x^3+x^4+...+x^9+x^{10})=1+sum_{10}=1+\frac{x*(x^{10}-1)}{x-1}$$.

(1) $$x<-1$$ --> $$1+\frac{x*(x^{10}-1)}{x-1}=1+\frac{negative*positive}{negative}=1+positive=positive$$. Sufficient.

(2) $$x^2>2$$ --> $$x<-\sqrt{2}$$ or $$x>\sqrt{2}$$ --> so, either $$1+\frac{x*(x^{10}-1)}{x-1}=1+\frac{negative*positive}{negative}=1+positive=positive$$ or $$1+\frac{x*(x^{10}-1)}{x-1}=1+\frac{positive*positive}{positive}=1+positive=positive$$. Sufficient.

Hope it's clear.
_________________
Intern
Joined: 20 Jul 2010
Posts: 5
Followers: 0

Kudos [?]: 0 [0], given: 7

### 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: 36598
Followers: 7095

Kudos [?]: 93457 [1] , given: 10563

### Show Tags

04 Jan 2011, 07:04
1
KUDOS
Expert's post
amankalra wrote:
Thanks!
I was actually considering a total of 11 terms, and b=1.
Is that right?

You can do that. If you take 1 as the first term then the formula will be $$sum=\frac{b*(r^{n}-1)}{r-1}=\frac{1*(x^{11}-1)}{x-1}=\frac{x^{11}-1}{x-1}$$.

(1) $$x<-1$$ --> $$\frac{x^{11}-1}{x-1}=\frac{negative}{negative}=positive$$. Sufficient.

(2) $$x^2>2$$ --> $$x<-\sqrt{2}\approx{-1.4}$$ or $$x>\sqrt{2}\approx{1.4}$$ --> so, either $$\frac{x^{11}-1}{x-1}=\frac{negative}{negative}=positive$$ or $$\frac{x^{11}-1}{x-1}=\frac{positive}{positive}=positive$$. Sufficient.

Hope it's clear.
_________________
Intern
Joined: 20 Jul 2010
Posts: 5
Followers: 0

Kudos [?]: 0 [0], given: 7

### Show Tags

04 Jan 2011, 07:08
Okay.
Thanks a ton!
Manager
Joined: 17 Dec 2010
Posts: 98
Location: Australia
GMAT 1: 690 Q48 V37
GPA: 3.37
WE: Engineering (Consulting)
Followers: 1

Kudos [?]: 23 [0], given: 15

### 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: 48
GMAT 1: 690 Q V
GPA: 3.7
Followers: 0

Kudos [?]: 188 [0], given: 117

### 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: 36598
Followers: 7095

Kudos [?]: 93457 [0], given: 10563

### Show Tags

25 Jan 2013, 05:34
alexpavlos wrote:
Is 1 + x + x^2 + … + x^10 positive?

1) x < -1
2) x^2 > 2

Merging similar topics. Please refer to the solutions above.
_________________
Current Student
Joined: 02 Jan 2013
Posts: 57
GMAT 1: 750 Q51 V40
GPA: 3.2
WE: Consulting (Consulting)
Followers: 0

Kudos [?]: 52 [1] , given: 2

### Show Tags

26 Jan 2013, 04:40
1
KUDOS
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)/(x-1) already mentioned.
Manager
Joined: 10 Mar 2014
Posts: 236
Followers: 1

Kudos [?]: 81 [0], given: 13

### 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.

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

Thanks
Director
Affiliations: GMATQuantum
Joined: 19 Apr 2009
Posts: 588
Followers: 112

Kudos [?]: 430 [0], given: 14

### 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^11-1)/(x-1)[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
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13501
Followers: 577

Kudos [?]: 163 [0], given: 0

### Show Tags

15 May 2015, 19:23
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 Legend
Joined: 09 Sep 2013
Posts: 13501
Followers: 577

Kudos [?]: 163 [0], given: 0

### Show Tags

18 Jun 2016, 11:30
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: Is 1+x+x^2+x^3+....+x^10 positive?   [#permalink] 18 Jun 2016, 11:30
Similar topics Replies Last post
Similar
Topics:
Is a positive? 2 09 Mar 2016, 11:51
2 Is x positive? 2 19 Aug 2015, 01:53
6 Is q positive? 4 14 Jan 2015, 01:29
17 Is x positive? 9 28 Jul 2012, 18:39
4 Is A positive? 44 08 Jun 2007, 22:45
Display posts from previous: Sort by