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Question

If $a_{0}, a_{1}, a_{2}, . . . .$ be the coefficients in the expansion of ${(1 + x + x^{2})}^{n}$ in ascending powers of $x$ , then $a_{0} a_{1} - a_{1} a_{2} + a_{2} a_{3} - . . . . =$

A

0

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B

1

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C

2

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D

- 1

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Solution

The correct option is C $0$
${(1 + x + x^{2})}^{n}$ $= a_{0} + a_{1} x + a_{2} x^{2} . . . . . + a_{n} x^{n}$
Let $n = 1$ in the above expression, thus as we can see from the expression,
the values of $a_{0} = 1$
also, $a_{1} = 1$
$a_{2} = 1$ and so on...
Thus, the value of
$a_{0} a_{1} - a_{1} a_{2} + a_{2} a_{3} - . . . . . . . . . . = 1 - 1 + 1 - 1 + 1.....$
Therefore, $a_{0} a_{1} - a_{1} a_{2} + a_{2} a_{3} - . . . . . . . . . . = 0$
Thus the correct answer is $0$

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