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Maths-

General

Easy

Question

If $left parenthesis 1 plus x plus x to the power of 2 end exponent right parenthesis to the power of n end exponent equals not stretchy sum from r equals 0 to 2 n of a subscript r end subscript x to the power of r end exponent comma blank t h e n blank a subscript 1 end subscript minus 2 a subscript 2 end subscript plus 3 a subscript 3 end subscript horizontal ellipsis minus 2 n a subscript 2 n end subscript$ is equal to

$n$
$- n$
0
$2 n$

The correct answer is: $negative n$

We have, $left parenthesis 1 plus x plus x to the power of 2 end exponent right parenthesis to the power of n end exponent equals a subscript 0 end subscript plus a subscript 1 end subscript x plus a subscript 2 end subscript x to the power of 2 end exponent plus a subscript 3 end subscript x to the power of 3 end exponent plus... plus a subscript 2 n end subscript x to the power of 2 n end exponent$
On differentiating both sides, we get
$n open parentheses 1 plus x plus x to the power of 2 end exponent close parentheses to the power of n minus 1 end exponent open parentheses 1 plus 2 x close parentheses equals a subscript 1 end subscript plus 2 a subscript 2 end subscript x plus 3 a subscript 3 end subscript x to the power of 2 end exponent plus... plus 2 n a subscript 2 n end subscript x to the power of 2 n minus 1 end exponent$
On putting $x equals negative 1$ , we get
$n open parentheses 1 minus 1 plus 1 close parentheses to the power of n minus 1 end exponent open parentheses 1 minus 2 close parentheses equals a subscript 1 end subscript minus 2 a subscript 2 end subscript plus 3 a subscript 3 end subscript minus... negative 2 n a subscript 2 n end subscript$
$rightwards double arrow blank a subscript 1 end subscript minus 2 a subscript 2 end subscript plus 3 a subscript 3 end subscript minus... negative 2 n a subscript 2 n end subscript equals negative n$

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