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  4. If y = 3x + 6x2 + 10x3 + . . . , then the value of x in terms of y is

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Q. If $ y = 3x + 6x^2 + 10x^3 + . . . $ , then the value of $ x $ in terms of $ y $ is

UPSEEUPSEE 2009

Solution:

Given, $y =3 x+6 x^{2}+10 x^{3}+\ldots$
$ \therefore 1+y=1+3 x+6 x^{2}+10 x^{3}+\ldots $
$\Rightarrow 1+y=(1-x)^{-3} $
$\Rightarrow 1-x=(1+y)^{-1 / 3} $
$\Rightarrow x=1-(1+y)^{-1 / 3}$