The coefficient of x^r in the expansion of fr | vedclass

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Question

(C) The given expression is $\frac{1}{\sqrt[3]{(1-2 x)^2}} = (1-2 x)^{-2/3}$.Using the binomial expansion $(1-z)^{-n} = 1 + nz + \frac{n(n+1)}{2!}z^2

Vedclass · Accepted Answer

$\frac{2 \cdot 5 \cdot 8 \ldots(3 r-1)}{r !}\left(\frac{2}{3}\right)^r$

Vedclass · Answer

(C) The given expression is $\frac{1}{\sqrt[3]{(1-2 x)^2}} = (1-2 x)^{-2/3}$.Using the binomial expansion $(1-z)^{-n} = 1 + nz + \frac{n(n+1)}{2!}z^2 + \dots + \frac{n(n+1)\dots(n+r-1)}{r!}z^r + \dots$ for $|z| The general term is $\frac{\frac{2}{3}(\frac{2}{3}+1)(\frac{2}{3}+2)\dots(\frac{2}{3}+r-1)}{r!}(2x)^r$.The coefficient of $x^r$ is $\frac{\frac{2}{3} \cdot \frac{5}{3} \cdot \frac{8}{3} \cdot \dots \cdot \frac{3r-1}{3}}{r!} \cdot 2^r$.$= \frac{2 \cdot 5 \cdot 8 \cdot \dots \cdot (3r-1)}{r! \cdot 3^r} \cdot 2^r$.$= \frac{2 \cdot 5 \cdot 8 \cdot \dots \cdot (3r-1)}{r!} \left(\frac{2}{3}\right)^r$.

The coefficient of $x^r$ in the expansion of $\frac{1}{\sqrt[3]{(1-2 x)^2}}$ is

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