The coefficient of $x^{48}$ in $(1 + x) + 2(1 + x)^{2} + 3(1 + x)^{3} + \ldots. + 100(1 + x)^{100}$

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The coefficient of $x^{48}$ in $(1 + x) + 2(1 + x)^{2} + 3(1 + x)^{3} + \ldots. + 100(1 + x)^{100}$ is equal to :

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Let $1 + x = r$$	herefore S = 1.r + 2.r^{2} + 3.r^{3} + \ldots\ldots + 100r^{100}\ldots\ldots$.$$\begin{array}{r} rS = 1.r^{2} + 2.r^{3} + \ldots.. + 99r^{100} + 100r^{101}\#(AGP) \end{array}$$(1) - (2) gives$$S = - \frac{(1 + x)^{101}}{x^{2}} + \frac{1}{x^{2}} + \frac{100(1 + x)^{101}}{x} $$∴ coefficient $x^{48}$ in S$= -$ coefficient of $x^{48}$ in $\frac{(1 + x)^{101}}{x^{2}} + 100$.Coefficient of $x^{48}$ in $\frac{(1 + x)^{101}}{x}$$$= 100\ ^{101}C_{49} - \ ^{101}C_{50}$$

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