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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 350,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink]
02 Jan 2011, 02:14

Expert's post

4

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.

Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink]
04 Jan 2011, 06:36

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 \(r=x\) --> \(1+(x+x^2+x^3+x^4+...+x^9+x^{10})=1+sum_{10}=1+\frac{x*(x^{10}-1)}{x-1}\).

(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.

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

1

This post received 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 \(sum=\frac{b*(r^{n}-1)}{r-1}=\frac{1*(x^{11}-1)}{x-1}=\frac{x^{11}-1}{x-1}\).

(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.

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

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

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

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

Michigan Ross: Center for Social Impact : The Center for Social Impact provides leaders with practical skills and insight to tackle complex social challenges and catalyze a career in...

The Importance of Financial Regulation : Before immersing in the technical details of valuing stocks, bonds, derivatives and companies, I always told my students that the financial system is...

The following pictures perfectly describe what I’ve been up to these days. MBA is an extremely valuable tool in your career, no doubt, just that it is also...