Prove that the coefficient of xn in the expansion 11-x-x2 is 1- 0 - -1 according as n is of the form 3m- 3m-1 or 3m -1

Let y=11+x+x^2 multiply and divide by 1 x y=1 x1 x^3=(1 x)[1 x^3]^ 1 expanding the later bracket =(1 x)[1+x^3+2.x^62!+2.3x^93! #160; . . . . ...

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Standard XII

Mathematics

Greatest Binomial Coefficients

Prove that th...

Question

Prove that the coefficient of $x^{n}$ in the expansion $\frac{1}{1 + x + x^{2}}$ is $1, 0, - 1$ according as $n$ is of the form $3 m, 3 m - 1$ or $3 m + 1$

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Solution

let
$y = \frac{1}{1 + x + x^{2}}$

multiply and divide by 1-x

$y = \frac{1 - x}{1 - x^{3}} = (1 - x) [1 - x^{3}]^{- 1}$

expanding the later bracket

$= (1 - x) [1 + x^{3} + 2. \frac{x^{6}}{2!} + 2.3 \frac{x^{9}}{3!} . . . . . . . (- 1) (- 2) . (- 3) . . (- m) \frac{(- x)^{3 m}}{m!} . . . . . . . . . . . . .]$

$= (1 - x) [1 + x^{3} + x^{6} + x^{9} . . . . . . x^{3 m - 3} + x^{3 m} . . . . . . . . . . . . .]$

coefficient of $x^{3 m}$
$= 1$

coefficient of $x^{3 m + 1}$
$= - 1$

coefficient of $x^{3 m - 1}$
$= 0$ because no such term exits

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Prove that the coefficient of xn in the expansion 11+x+x2 is 1,0,−1 according as n is of the form 3m,3m−1 or 3m+1

Prove that the coefficient of $x^{n}$ in the expansion $\frac{1}{1 + x + x^{2}}$ is $1, 0, - 1$ according as $n$ is of the form $3 m, 3 m - 1$ or $3 m + 1$