The coefficientof x^{101} in the expression (5 + x)... | Sequence & Series (Mathematics)

The coefficientof

x^{101}

in the expression

(5 + x)^{500} + x(5 + x)^{499} + x^{2}(5 + x)^{498} + \ldots + x^{500},x > 0

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

\ ^{501}C_{101}(5)^{400}

\ ^{501}C_{100}(5)^{400}

\ ^{500}C_{101}(5)^{399}

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Correct Answer:

Explanation

(a) : The given series is a G.P. with $a = (5 + x)^{500}\\$ $r = \frac{x}{5 + x}\\$ $\therefore$ Sum of G.P. $= (5 + x)^{500}\left\{ \frac{1 - \left( \frac{x}{5 + x} \right)^{501}}{1 - \frac{x}{5 + x}} \right\}$ $\\= \frac{\left( (5 + x)^{501} - x^{501} \right)}{5}\\$ Now, coefficient of $x^{101}$ in $\frac{1}{5}\left\lbrack (5 + x)^{501} - x^{501} \right\rbrack$ $\\= \frac{1}{5} ^{501}C_{101}(5)^{400} \\= ^{501}C_{101} 5^{399}$

Explanation

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