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# M19-05

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Math Expert
Joined: 02 Sep 2009
Posts: 43321

Kudos [?]: 139486 [0], given: 12790

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16 Sep 2014, 00:05
Expert's post
6
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BOOKMARKED
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Difficulty:

35% (medium)

Question Stats:

66% (01:00) correct 34% (01:00) wrong based on 161 sessions

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Is $$1 + x + x^2 + ... + x^{10}$$ positive?

(1) $$x \lt -1$$

(2) $$x^2 \gt 2$$
[Reveal] Spoiler: OA

_________________

Kudos [?]: 139486 [0], given: 12790

Math Expert
Joined: 02 Sep 2009
Posts: 43321

Kudos [?]: 139486 [0], given: 12790

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16 Sep 2014, 00:05
Expert's post
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Official Solution:

Is $$1+(x+x^2)+(x^3+x^4)+...+(x^9+x^{10}) \gt 0$$?

(1) $$x \lt -1$$. From this statement it follows that $$x+x^2 \gt 0$$, $$x^3+x^4 \gt 0$$, ..., $$x^9+x^{10} \gt 0$$, so the sum is also more than zero. Sufficient.

(2) $$x^2 \gt 2$$. This statement implies that $$x \lt -\sqrt{2}$$ or $$x \gt \sqrt{2}$$. Even if $$x$$ itself is negative then still as above: $$x+x^2 \gt 0$$, $$x^3+x^4 \gt 0$$, ..., $$x^9+x^{10} \gt 0$$, so the sum is also more than zero. Sufficient.

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26 Mar 2017, 08:38
Bunuel wrote:
Is $$1 + x + x^2 + ... + x^{10}$$ positive?

(1) $$x \lt -1$$

(2) $$x^2 \gt 2$$

So essentially we need to figure whether x is between 0-1 or not. is that correct? Thank you.

Kudos [?]: 31 [0], given: 510

Intern
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02 Apr 2017, 15:20
Strange question...
Wouldn't the answer be positive regardless what the statement is?

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04 Apr 2017, 07:35
1
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Expert's post
Bunuel wrote:
Is $$1 + x + x^2 + ... + x^{10}$$ positive?

(1) $$x \lt -1$$

(2) $$x^2 \gt 2$$

Hi,

The Q in the present format may not require any statement.

$$1 + x + x^2 + ... + x^{10}$$

1) If x is POSITIVE, it will always be positive.

2) If x is NEGATIVE and x is between 0 and -1.
$$1 + x + x^2 + ... + x^{10}= (1+x)+(x^2+x^3)+....(x^8+x^9)+x^{10}$$
Here x is between 0 and 1..
So 1+x will be positive.
$$x^2+x^3$$ will have x^2 as positive and x^3 as negative but the numeric value of lower powers will be greater. Example (1/2)^2=1/4 whereas (1/2)^3 is 1/8....
Similarly all other brackets too would be positive and thus total equation will be positive..
3) If x is NEGATIVE and <-1..
$$1 + x + x^2 + ... + x^{10}= 1+(x+x^2)+(x^3+x^4)+.....+(x^9+x^{10})$$..
Here each bracket will be POSITIVE as higher power mean higher value and each bracket has higher EVEN power.
Again overall POSITIVE.

Bunuel, the Q will be correct if we remove 1 from equation and Q becomes
Is $$x + x^2 + ... + x^{10}$$ positive?
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

BANGALORE/-

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Math Expert
Joined: 02 Sep 2009
Posts: 43321

Kudos [?]: 139486 [0], given: 12790

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04 Apr 2017, 07:43
chetan2u wrote:
Bunuel wrote:
Is $$1 + x + x^2 + ... + x^{10}$$ positive?

(1) $$x \lt -1$$

(2) $$x^2 \gt 2$$

Hi,

The Q in the present format may not require any statement.

$$1 + x + x^2 + ... + x^{10}$$

1) If x is POSITIVE, it will always be positive.

2) If x is NEGATIVE and x is between 0 and -1.
$$1 + x + x^2 + ... + x^{10}= (1+x)+(x^2+x^3)+....(x^8+x^9)+x^{10}$$
Here x is between 0 and 1..
So 1+x will be positive.
$$x^2+x^3$$ will have x^2 as positive and x^3 as negative but the numeric value of lower powers will be greater. Example (1/2)^2=1/4 whereas (1/2)^3 is 1/8....
Similarly all other brackets too would be positive and thus total equation will be positive..
3) If x is NEGATIVE and <-1..
$$1 + x + x^2 + ... + x^{10}= 1+(x+x^2)+(x^3+x^4)+.....+(x^9+x^{10})$$..
Here each bracket will be POSITIVE as higher power mean higher value and each bracket has higher EVEN power.
Again overall POSITIVE.

Bunuel, the Q will be correct if we remove 1 from equation and Q becomes
Is $$x + x^2 + ... + x^{10}$$ positive?

Thank you. I'm revising the question.
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06 Apr 2017, 08:23
If I'm understanding this correctly, you just need to know if x is between 0 and -1.

Posted from my mobile device

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Manager
Joined: 14 Jun 2016
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31 Jul 2017, 03:46
Bunuel wrote:
chetan2u wrote:
Bunuel wrote:
Is $$1 + x + x^2 + ... + x^{10}$$ positive?

(1) $$x \lt -1$$

(2) $$x^2 \gt 2$$

Hi,

The Q in the present format may not require any statement.

$$1 + x + x^2 + ... + x^{10}$$

1) If x is POSITIVE, it will always be positive.

2) If x is NEGATIVE and x is between 0 and -1.
$$1 + x + x^2 + ... + x^{10}= (1+x)+(x^2+x^3)+....(x^8+x^9)+x^{10}$$
Here x is between 0 and 1..
So 1+x will be positive.
$$x^2+x^3$$ will have x^2 as positive and x^3 as negative but the numeric value of lower powers will be greater. Example (1/2)^2=1/4 whereas (1/2)^3 is 1/8....
Similarly all other brackets too would be positive and thus total equation will be positive..
3) If x is NEGATIVE and <-1..
$$1 + x + x^2 + ... + x^{10}= 1+(x+x^2)+(x^3+x^4)+.....+(x^9+x^{10})$$..
Here each bracket will be POSITIVE as higher power mean higher value and each bracket has higher EVEN power.
Again overall POSITIVE.

Bunuel, the Q will be correct if we remove 1 from equation and Q becomes
Is $$x + x^2 + ... + x^{10}$$ positive?

Thank you. I'm revising the question.

The question is still not reversed; please remove 1...thanks
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19 Aug 2017, 23:12
it is a GP
1+X^2+X^3....+X^10= (X^11-1)/(X-1)
RHS is always positive if |X|>1

1- Statement: 1..says X<-1, means |X|>1...therefore RHS is positive always: Statement is sufficient
2- Statement: 2..says X^2>2, means |X|>1...therefore RHS is positive always: Statement is sufficient

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Re: M19-05   [#permalink] 19 Aug 2017, 23:12
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# M19-05

Moderators: chetan2u, Bunuel

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