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If $x\in R,$ then the value of $S=1-^nC_1.\large\frac{1+x}{1+nx}$$+^nC_2.\large\frac{1+2x}{(1+nx)^2}$$-^nC_3.\large\frac{1+3x}{(1+nx)^3}$$+..............(n+1) $terms = ?

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Let $\large\frac{1}{1+nx}=y$
Then $S=\big( ^nC_0-^nC_1y$$+ ^nC_2y^2-....$$(-1)^n$$ ^nC_n $$y^n\big)-$
$xy\big(^nC_1-2.^nC_2y+3.^nC_2y^2-.........(-1)^{n-1}$$n.^nC_ny^{n-1}\big)$
$=(1-y)^n+xy.\large\frac{\delta}{\delta y}$${(1-y)^n}$
$=(1-y)^n+nxy(1-y)^{n-1}$
$=(1-y)^{n-1}.\big[(1-y)+nxy\big]$
$=(1-y)^{n-1}.0=0$   $\big($ by substituting the value of $y=\large\frac{1}{1+nx}\big)$

 

answered Oct 22, 2013 by rvidyagovindarajan_1
edited Oct 22, 2013 by rvidyagovindarajan_1
 

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