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Algebra Level 3

Find the value of

x 2014 + x 2013 + x 2012 + . . . + x 3 + x 2 + x 1 x 2014 + 1 x 2013 + 1 x 2012 + . . . + 1 x 3 + 1 x 2 + 1 x \frac {x^{2014} + x^{2013} + x^{2012} + . . . + x^3 + x^2 + x} { \frac {1}{x^{2014}} + \frac {1}{x^{2013}} + \frac {1}{x^{2012}} + . . . + \frac {1}{x^3} + \frac {1}{x^2} + \frac {1}{x}}

x^2 x^2014 1 x^2015

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3 solutions

First, simplify the denominator.

x 2014 + x 2013 + . . . + x 2 + x 1 1 + x + x 2 + . . . + x 2013 x 2014 \frac {x^{2014} + x^{2013} + . . . + x^2 + x^1}{ \frac{1 + x + x^2 + . . . + x^{2013}}{x^{2014}} }

factor out one x in the numerator, then reciprocate the denominator to make solution simplier

x ( x 2013 + . . . + x 2 + x 1 + 1 ) x 2014 1 + x + x 2 + . . . + x 2013 x( x^{2013} + . . . + x^2 + x^1 + 1) \cdot \frac {x^{2014}} {1 + x + x^2 + . . . + x^{2013}}

you can now cancel the long part of the solution,

x x 2014 = x 2015 x \cdot x^{2014} = x^{2015}

didn't get it..... Please explain in some other way in more simplistic way please

Abhishek Chawla - 7 years, 2 months ago

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sorry for the inconveniece, it is a hard time for me to show simple solution because i dont know how to use latex.

Hans Lawrence Dela Cruz - 7 years, 2 months ago

i think i got a better solution :)

Hans Lawrence Dela Cruz - 7 years, 2 months ago

please go through the formula of calculating sum of a G.P. series ... u will get it

Arijit Banerjee - 7 years, 1 month ago

i just try this with 2^1 + 2^2 + 2^3 : 1/2^1 + 1/2^2 + 1/2^3 and the result is 2^4 ._. so the result of that question is x^2015 :v

try for x^2 :U

math man - 6 years, 11 months ago
Arijit Banerjee
Apr 15, 2014

Apply formula for calculating sum of a G.P series.... then simplify it answer will come

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