Is 1+x+x^2+x^3+....+x^10 positive? 1. x<-1 2. x^2>2

Oldest

Best Reply

Author

Message

TAGS:

Senior Manager

Status: ready to boMBArd

Joined: 31 Oct 2010

Posts: 493

Location: India

Concentration: Entrepreneurship, Strategy

Schools: Kellogg 1YR '13 (S), Johnson '14 (WL), ISB (D), Carlson (A)

GMAT 1: 710 Q48 V40

WE: Project Management (Manufacturing)

Followers: 10

Kudos [?]: 31 [0], given: 67

Is 1+x+x^2+x^3+....+x^10 positive? 1. x<-1 2. x^2>2 [#permalink]

02 Jan 2011, 02:34

00:00

Question Stats:

66% (01:59) correct

33% (01:13) wrong

based on 0 sessions

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

1. x<-1
2. x^2>2

Source: GMAT Club Hardest problems.

[Reveal] Spoiler: OA

_________________

My GMAT debrief: from-620-to-710-my-gmat-journey-114437.html

Kaplan GMAT Prep Discount Codes

Knewton GMAT Discount Codes

Manhattan GMAT Discount Codes

GMAT Club team member

Joined: 02 Sep 2009

Posts: 11540

Followers: 1795

Kudos [?]: 9567 [0], given: 826

Re: Is the sum positive? [#permalink]

02 Jan 2011, 03:14

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.
_________________

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

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. NEW!!!

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. NEW!!!

What are GMAT Club Tests?
25 extra-hard Quant Tests

Find out what's new at GMAT Club - latest features and updates

Senior Manager

Status: ready to boMBArd

Joined: 31 Oct 2010

Posts: 493

Location: India

Concentration: Entrepreneurship, Strategy

Schools: Kellogg 1YR '13 (S), Johnson '14 (WL), ISB (D), Carlson (A)

GMAT 1: 710 Q48 V40

WE: Project Management (Manufacturing)

Followers: 10

Kudos [?]: 31 [0], given: 67

Re: Is the sum positive? [#permalink]

03 Jan 2011, 11:34

Its clear now! Thanks!!
_________________

My GMAT debrief: from-620-to-710-my-gmat-journey-114437.html

Intern

Joined: 20 Jul 2010

Posts: 5

Followers: 0

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

Re: Is the sum positive? [#permalink]

04 Jan 2011, 06:56

Hi!
This is a geometric progression. Could you explain this using the sum formula?

Thanks!

GMAT Club team member

Joined: 02 Sep 2009

Posts: 11540

Followers: 1795

Kudos [?]: 9567 [0], given: 826

Re: Is the sum positive? [#permalink]

04 Jan 2011, 07:36

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)}{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.

Answer: D.

Hope it's clear.
_________________

Intern

Joined: 20 Jul 2010

Posts: 5

Followers: 0

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

Re: Is the sum positive? [#permalink]

04 Jan 2011, 07:54

Thanks!
I was actually considering a total of 11 terms, and b=1.
Is that right?

GMAT Club team member

Joined: 02 Sep 2009

Posts: 11540

Followers: 1795

Kudos [?]: 9567 [1] , given: 826

Re: Is the sum positive? [#permalink]

04 Jan 2011, 08:04

This post received
KUDOS

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.

Answer: D.

Hope it's clear.
_________________

Intern

Joined: 20 Jul 2010

Posts: 5

Followers: 0

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

Re: Is the sum positive? [#permalink]

04 Jan 2011, 08:08

Okay.
Thanks a ton!

Manager

Joined: 17 Dec 2010

Posts: 100

Location: Australia

GMAT 1: 690 Q48 V37

GPA: 3.37

WE: Engineering (Consulting)

Followers: 1

Kudos [?]: 17 [0], given: 15

Re: Is the sum positive? [#permalink]

04 Jan 2011, 21:14

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

Manager

Joined: 18 Mar 2012

Posts: 50

Followers: 0

Kudos [?]: 2 [0], given: 117

is 1 + x + x^2 + … + x^10 positive [#permalink]

25 Jan 2013, 06:05

Is 1 + x + x^2 + … + x^10 positive?

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

GMAT Club team member

Joined: 02 Sep 2009

Posts: 11540

Followers: 1795

Kudos [?]: 9567 [0], given: 826

Re: is 1 + x + x^2 + … + x^10 positive [#permalink]

25 Jan 2013, 06:34

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.
_________________

Manager

Joined: 02 Jan 2013

Posts: 50

Followers: 0

Kudos [?]: 31 [0], given: 1

Re: Is 1+x+x^2+x^3+....+x^10 positive? 1. x<-1 2. x^2>2 [#permalink]

26 Jan 2013, 05:40

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)/(x-1) already mentioned.
_________________

Please press "kudo" if this helped you!

Re: Is 1+x+x^2+x^3+....+x^10 positive? 1. x<-1 2. x^2>2 [#permalink] 26 Jan 2013, 05:40

	Similar topics	Author	Replies	Last post
Similar Topics:	If x?0, is \|x\| <1? (1) x^2<1 (2) \|x\| < 1/x	MA	10	09 Mar 2005, 00:20
	Is 1 + x + (x^2) + ...+ (x^10) positive? 1. x <1> 2	bmwhype2	1	14 Nov 2007, 03:51
	If x≠0, is \|x\| <1? (1) x^2 <1 (2) \|x\| < 1/x	SMAbbas	6	19 Oct 2009, 07:10
	If x 0, is x^2 / \|x\| < 1?	zaarathelab	7	14 Dec 2009, 01:03
	If x 0, is \|x\| <1? (1) x^2<1 (2) \|x\| < 1/2	banksy	3	26 Feb 2011, 16:23

Is 1+x+x^2+x^3+....+x^10 positive? 1. x<-1 2. x^2>2

Main navigation

GMAT Resources

Partners

MBA Resources

GMAT ® is a registered trademark of the Graduate Management Admission Council ® (GMAC ®). GMAT Club's website has not been reviewed or endorsed by GMAC.

GMAT Courses

Admissions Consulting

Is 1+x+x^2+x^3+....+x^10 positive? 1. x<-1 2. x^2>2

Is 1+x+x^2+x^3+....+x^10 positive? 1. x<-1 2. x^2>2

Main navigation

GMAT Resources

Partners

MBA Resources