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What is the value of x1*x2*...*x100 ?

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Bunuel
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What is the value of x1*x2*...*x100 ? [#permalink] New post   01 Sep 2022, 02:26
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00:00
A
B
C
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E

Difficulty:

35% (medium)

Question Stats:

68% (01:39) correct 32% (01:41) wrong based on 117 sessions
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\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
...
\(x_1= 100\).

What is the value of \(x_1*x_2*...*x_{100} \)?

A. -100
B. -1
C. 0
D. 1
E. 100

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A
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chetan2u
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Re: What is the value of x1*x2*...*x100 ? [#permalink] New post   01 Sep 2022, 02:47
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Bunuel wrote:
\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
...
\(x_1= 100\).

What is the value of \(x_1*x_2*...*x_{100} \)?

A. -100
B. -1
C. 0
D. 1
E. 100

Fresh GMAT Club Tests' Question



\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
So \(2+x_{100}=1……x_{100}=-1\)

\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
So, \(3+x_{98}=2………x_{98}=-1\)
...

and so on till \(x_2=1\)
Also, \(x_1= 100\).
Thus, \(x_1*x_2*...*x_{100}=100*-1*-1….*-1=100*(-1)^{99}=100*(-1)=-100 \)



A
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Rabab36
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Re: What is the value of x1*x2*...*x100 ? [#permalink] New post   01 Sep 2022, 02:54
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x1+x2+...+x100=1x1+x2+...+x100=1; -------A
x1+x2+...+x99=2x1+x2+...+x99=2;--------B
x1+x2+...+x98=3x1+x2+...+x98=3;---------C

NOW A-B
GIVES YOU X100=-1
B-C GIVES YOU X99=-1
SO ON.....

ALSO GIVEN X1=100

THEREFORE
100 *(-1)(-1)......(-1)
SINCE -1 ARE ODD NO OF TERMS

THEREFORE -100

CORRECT OPTION A
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harshpathak96
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Re: What is the value of x1*x2*...*x100 ? [#permalink] New post   12 Oct 2022, 23:42
chetan2u wrote:
Bunuel wrote:
\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
...
\(x_1= 100\).

What is the value of \(x_1*x_2*...*x_{100} \)?

A. -100
B. -1
C. 0
D. 1
E. 100

Fresh GMAT Club Tests' Question



\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
So \(2+x_{100}=1……x_{100}=-1\)

\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
So, \(3+x_{98}=2………x_{98}=-1\)
...

and so on till \(x_2=1\)
Also, \(x_1= 100\).
Thus, \(x_1*x_2*...*x_{100}=100*-1*-1….*-1=100*(-1)^{99}=100*(-1)=-100 \)



A



I think there is a typo, it should be X2 = -1, right?
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Bunuel
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Re: What is the value of x1*x2*...*x100 ? [#permalink] New post   11 Mar 2023, 03:13
Expert Reply
Bunuel wrote:
\(x_1+x_2+...+x_{100} = 1\);
\(x_1+x_2+...+x_{99} = 2\);
\(x_1+x_2+...+x_{98} = 3\);
...
\(x_1= 100\).

What is the value of \(x_1*x_2*...*x_{100} \)?

A. -100
B. -1
C. 0
D. 1
E. 100

Fresh GMAT Club Tests' Question


Official Solution:


\(x_1+x_2+...+x_{100} = 1\);

\(x_1+x_2+...+x_{99} = 2\);

\(x_1+x_2+...+x_{98} = 3\);

\(...\)

\(x_1= 100\).

What is the value of \(x_1*x_2*...*x_{100}\)?


A. \(-100\)
B. \(-1\)
C. \(0\)
D. \(1\)
E. \(100\)


By subtracting the second equation from the first, we obtain \((x_1+x_2+...+x_{100})-(x_1+x_2+...+x_{99}) = 1-2\), which simplifies to \(x_{100}= -1\).

Similarly, subtracting the third equation from the second yields \((x_1+x_2+...+x_{99}) -(x_1+x_2+...+x_{98}) = 2-3\), which simplifies to \(x_{99}= -1\).

By repeating this process for each successive pair of equations, we can determine that every term up to \(x_1\) is also equal to -1.

Therefore, \(x_1*x_2*...*x_{100}=100(-1)(-1)(-1)...(-1)=100(-1)^{99}=-100\).


Answer: A
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Re: What is the value of x1*x2*...*x100 ? [#permalink]
11 Mar 2023, 03:13
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