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

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Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 02 Jan 2011, 01:34
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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

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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 02 Jan 2011, 02:14
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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.
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 03 Jan 2011, 10:34
Its clear now! Thanks!!
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 04 Jan 2011, 05:56
Hi!
This is a geometric progression. Could you explain this using the sum formula?

Thanks!
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 04 Jan 2011, 06:36
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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.
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 04 Jan 2011, 06:54
Thanks!
I was actually considering a total of 11 terms, and b=1.
Is that right?
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 04 Jan 2011, 07:04
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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.

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


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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 25 Jan 2013, 05:05
Is 1 + x + x^2 + … + x^10 positive?

1) x < -1
2) x^2 > 2
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 25 Jan 2013, 05:34
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 26 Jan 2013, 04:40
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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.
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 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
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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 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.

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Re: Is 1+x+x^2+x^3+....+x^10 positive? [#permalink] New post 15 May 2015, 19:23
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Re: Is 1+x+x^2+x^3+....+x^10 positive?   [#permalink] 15 May 2015, 19:23
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