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Math Expert V
Joined: 02 Sep 2009
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5 00:00

Difficulty:   45% (medium)

Question Stats: 61% (01:30) correct 39% (01:41) wrong based on 170 sessions

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

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

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

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Math Expert V
Joined: 02 Sep 2009
Posts: 55631

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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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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.
Current Student B
Joined: 28 Dec 2016
Posts: 88
Location: United States (IL)
Concentration: Marketing, General Management
Schools: Johnson '20 (M)
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Strange question...
Wouldn't the answer be positive regardless what the statement is?
Math Expert V
Joined: 02 Aug 2009
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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?
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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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If I'm understanding this correctly, you just need to know if x is between 0 and -1.

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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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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

Intern  B
Joined: 02 Feb 2018
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buan15 wrote:

The question is still not reversed; please remove 1...thanks

+1
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barryseal wrote:
buan15 wrote:

The question is still not reversed; please remove 1...thanks

+1

what does that mean?
Intern  B
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prabsahi wrote:
barryseal wrote:
buan15 wrote:

The question is still not reversed; please remove 1...thanks

+1

what does that mean?

that question is still not adjusted
Intern  B
Joined: 07 Dec 2017
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Hi!!

The question is: 1+x+x2+...+x10 positive?

Why do you assume that the set is 1+x+x2+x3+x4+x5+x6+x7+x8+x9+x10?

Since the question stem dosen't say anything about the nature of the set, you can form many different sets such as
1+x+x2+x2+x3+x3+x3+......+x10
So, we can have several answers Re: M19-05   [#permalink] 14 Apr 2019, 06:31
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