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What is the value of the positive root of 1x-5(x-4)+1x-4(x-3)+…+1x+4(x+5)=512?

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by Max@Math Revolution » Fri May 15, 2020 6:27 am

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[GMAT math practice question]

What is the value of the positive root of 1/(x-5)(x-4)+1/(x-4)(x-3)+…+1/(x+4)(x+5)=5/12?

A. 3
B. 5
C. 7
D. 9
E. 11
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Re: What is the value of the positive root of 1x-5(x-4)+1x-4(x-3)+…+1x+4(x+5)=512?

by Max@Math Revolution » Sun May 17, 2020 6:28 pm
=>

Remember that 1/A(A+1)=1/A-1/A+1.
1/(x-5)(x-4)=1/x-5-1/x-4
1/(x-4)(x-3)=1/x-4-1/x-3
……
1/(x+4)(x+5)=1/x-5-1/x-4

1/(x-5)(x-4)+1/(x-4)(x-3)+…+1/(x+4)(x+5)
=1/x-5-1/x-4+1/x-4-1/x-3+…+1/x+4-1/x+5
=1/x-5-1/x+5
=10/(x-5)(x+5)=5/12
We have 5(x - 5)(x + 5) = 120, (x - 5)(x + 5) = 24, or x^2 = 49.

Then we have x = 7 since x is positive.

Therefore, C is the answer.
Answer: C
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