The coefficient of x^{101} in the expression | vedclass

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Question

(A) The given expression is a geometric series with first term $a = (5+x)^{500}$,common ratio $r = \frac{x}{5+x}$,and number of terms $n = 501$.The su

Vedclass · Accepted Answer

$^{501}C_{101}(5)^{399}$

Vedclass · Answer

(A) The given expression is a geometric series with first term $a = (5+x)^{500}$,common ratio $r = \frac{x}{5+x}$,and number of terms $n = 501$.The sum of a geometric series is given by $S_n = \frac{a(1-r^n)}{1-r}$.Substituting the values:$S = \frac{(5+x)^{500} \left[ 1 - \left( \frac{x}{5+x} ight)^{501} ight]}{1 - \frac{x}{5+x}}$$S = \frac{(5+x)^{500} \left[ \frac{(5+x)^{501} - x^{501}}{(5+x)^{501}} ight]}{\frac{5+x-x}{5+x}}$$S = \frac{(5+x)^{501} - x^{501}}{5+x} 	imes \frac{5+x}{5} = \frac{(5+x)^{501} - x^{501}}{5}$We need the coefficient of $x^{101}$ in $\frac{1}{5} [(5+x)^{501} - x^{501}]$.The term containing $x^{101}$ in $(5+x)^{501}$ is given by the binomial expansion: $^{501}C_{101} (5)^{501-101} x^{101} = ^{501}C_{101} (5)^{400} x^{101}$.Therefore,the coefficient of $x^{101}$ in the expression is $\frac{1}{5} 	imes ^{501}C_{101} (5)^{400} = ^{501}C_{101} (5)^{399}$.

The coefficient of $x^{101}$ in the expression $(5+x)^{500} + x(5+x)^{499} + x^{2}(5+x)^{498} + \ldots + x^{500}$ for $x > 0$ is:

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