The coefficient of xn in the expansion of 1-x-2x31-x3 is

The correct option is A n 1+x 2 x ^ 3 ( 1 x ) #160; ^ 3 = 1+x x ^ 3 x ^ 3 ( 1 x ) #160; ^ 3 = 1 x ^ 3 +x x ^ 3 ( 1 x ) #160; ^ 3 ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

The coefficie...

Question

The coefficient of $x^{n}$ in the expansion of $\frac{1 + x - 2 x^{3}}{(1 - x)^{3}}$ is

n

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3 n - 1

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4 n - 1

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5 n - 1

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Solution

The correct option is A $n$
$\frac{1 + x - 2 x^{3}}{{(1 - x)}^{3}}$
$= \frac{1 + x - x^{3} - x^{3}}{{(1 - x)}^{3}}$
$= \frac{1 - x^{3} + x - x^{3}}{{(1 - x)}^{3}}$
$= \frac{(1 - x^{3}) + x (1 - x^{2})}{{(1 - x)}^{3}}$
$= \frac{(1 - x) (1 + x^{2} + x) + x (1 - x) (1 + x)}{{(1 - x)}^{3}}$
$= \frac{(1 - x) [1 + x^{2} + x + x + x^{2}]}{{(1 - x)}^{3}}$
$= \frac{2 x^{2} + 2 x + 1}{{(1 - x)}^{2}}$
$= (2 x^{2} + 2 x + 1) {(1 - x)}^{- 2}$
${(1 - x)}^{- 2} = 1 - (- 2) x - \frac{(- 2) (- 3)}{2} x^{2} + . . . . .$
$= 1 + 2 x - 3 x^{2} + . . . .$
$∴ \frac{1 + x - 2 x^{3}}{{(1 - x)}^{3}}$
$= (2 x^{2} + 2 x + 1) (1 + 2 x - 3 x^{2} + . . .)$
Coeff. of $x^{0} = 1$
Coeff. of $x = 4$
Coeff. of $x^{2} = 3$
Coeff. of $x^{3} = 2$

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The coefficient of xn in the expansion of 1+x−2x3(1−x)3 is

The coefficient of $x^{n}$ in the expansion of $\frac{1 + x - 2 x^{3}}{(1 - x)^{3}}$ is