Prove that the coefficient of xr in the expansion of 1-4x- 1-2 in -2r-r-2

(1+x)^n=1+nx+(n)(n 1)x^22!+ . . . . . . (n!)x^nn! #10; #10;Where n is negative or fractiongiven that (1 4x)^ 12=1+( 12)x+( 12)( 12 1)x^22!+ . . . ...

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Standard IX

Mathematics

Linear Equation in One Variable

Prove that th...

Question

Prove that the coefficient of $x^{r}$ in the expansion of $(1 - 4 x)^{- \frac{1}{2}}$ in $\frac{| 2 r - - - |}{(| r - |)^{2}}$

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Solution

$(1 + x)^{n} = 1 + n x + (n) (n - 1) \frac{x^{2}}{2!} + . . . . . . (n!) \frac{x^{n}}{n!}$

Where n is negative or fraction
given that
$(1 - 4 x)^{\frac{- 1}{2}} = 1 + (\frac{- 1}{2}) x + (\frac{- 1}{2}) (\frac{- 1}{2} - 1) \frac{x^{2}}{2!} + . . . . . . (\frac{- 1}{2}) (\frac{- 1}{2} - 1) . . . (\frac{- 1}{2} - r + 1) \frac{x^{r}}{r!}$

coefficient of $x^{r}$ is
$\frac{(- 1) . (- 3) . (- 5) . . . . (1 - 2 r)}{2^{r} . r!}$

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Prove that the coefficient of xr in the expansion of (1−4x)−12 in |2r–––|(|r–|)2

(1+x)n=1+nx+(n)(n−1)x22!+......(n!)xnn! Where n is negative or fractiongiven that (1−4x)−12=1+(−12)x+(−12)(−12−1)x22!+......(−12)(−12−1)...(−12−r+1)xrr!coefficient of xr is(−1).(−3).(−5)....(1−2r)2r.r!

Prove that the coefficient of $x^{r}$ in the expansion of $(1 - 4 x)^{- \frac{1}{2}}$ in $\frac{| 2 r - - - |}{(| r - |)^{2}}$