If $a_{0}, a_{1}, a_{2}, a_{3} . . . .$ are the coefficient in order of the expansion of $(1 + x + x^{2})^{n}$ , prove that
$a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + . . . . . + (- 1)^{n - 1} a_{n - 1}^{2} = \frac{1}{2} a_{n} {1 - (- 1)^{n} a_{n}}$ .

Open in App

Solution

${(1 + x + x^{2})}^{n} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{n} x^{n} + . . . + a_{3} x^{2 n - 3} + a_{2} x^{2 n - 2} + a_{1} x^{2 n - 1} + a_{0} x^{2 n}$

$a_{n}$ comes once in the series.

Replace $x$ by $- x$ ,

${(1 - x + x^{2})}^{n} = a_{0} - a_{1} x + a_{2} x^{2} + . . . + {(- 1)}^{n} a_{n} x^{n} + . . . - a_{3} x^{2 n - 3} + a_{2} x^{2 n - 2} - a_{1} x^{2 n - 1} + a_{0} x^{2 n}$

Multiplying the both series and get the coefficient of $x^{2 n}$ is ,
$2 a_{0}^{2} - 2 a_{1}^{2} + 2 a_{2}^{2} - 2 a_{3}^{2} . . . + 2 {(- 1)}^{n - 2} a_{n - 2}^{2} + 2 {(- 1)}^{n - 1} a_{n - 1}^{2} + {(- 1)}^{n} a_{n}^{2}$

Coefficient of $x^{2 n}$ in this ${(1 + x + x^{2})}^{n} {(1 - x + x^{2})}^{n}$ will be $a_{n}$

$a_{n} = 2 a_{0}^{2} - 2 a_{1}^{2} + 2 a_{2}^{2} - 2 a_{3}^{2} . . . + 2 {(- 1)}^{n - 2} a_{n - 2}^{2} + 2 {(- 1)}^{n - 1} a_{n - 1}^{2} + {(- 1)}^{n} a_{n}$

$a_{n} = 2 (a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} . . . + {(- 1)}^{n - 2} a_{n - 2}^{2} + {(- 1)}^{n - 1} a_{n - 1}^{2}) + {(- 1)}^{n} a_{n}^{2}$

$a_{n} - {(- 1)}^{n} a_{n}^{2} = 2 (a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} . . . + {(- 1)}^{n - 2} a_{n - 2}^{2} + {(- 1)}^{n - 1} a_{n - 1}^{2})$

$\frac{a_{n}}{2} (1 - {(- 1)}^{n} a_{n}) = a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} . . . + {(- 1)}^{n - 2} a_{n - 2}^{2} + {(- 1)}^{n - 1} a_{n - 1}^{2}$

$a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} . . . + {(- 1)}^{n - 2} a_{n - 2}^{2} + {(- 1)}^{n - 1} a_{n - 1}^{2} = \frac{a_{n}}{2} (1 - {(- 1)}^{n} a_{n})$

Suggest Corrections

Similar questions

a_{0}, a_{1}, a_{2}

^{2})^{n}

a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + . . . . . . + (- 1)^{n - 1} a_{n - 1}^{2}

= \frac{1}{2} a_{n} (1 - (- 1)^{n} a_{n})

a_{0}, a_{1}, a_{2}, . . . . . .

(1 + x + x^{2})^{n}

x

a_{0}^{2} - a_{1}^{2} + a_{2}^{2} - a_{3}^{2} + . . . . . . + a_{2 a}^{2} = a_{n}

a_{0}, a_{1}, a_{2}, . . . . .

(1 + x + x^{2})^{n}

(a_{0} + a_{3} + a_{6} + . . .) = (a_{1} + a_{4} + a_{7} + . . . .) = = (a_{2} + a_{5} + a_{8} + . . .) = 3^{n - 1}

a_{0}, a_{1}, a_{2}, . . . .

{(1 + x + x^{2})}^{n}

x

a_{0} a_{1} - a_{1} a_{2} + a_{2} a_{3} - . . . . =

a_{1}, a_{2}, a_{3}, a_{4}

a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1

(a_{1} - a_{2})^{2} + (a_{2} - a_{2})^{2} + (a_{3} - a_{4})^{2} + (a_{4} - a_{1})^{2}

Join BYJU'S Learning Program