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Higher Order Derivatives

If 1+x1+x21+x...

Question

If $(1 + x) (1 + x^{2}) (1 + x^{4}) . . . (1 + x^{128}) = n \sum r = 0 x^{r},$ then n is equal to

A

256

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B

225

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C

254

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D

none of these

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Solution

The correct option is B
225

Degree of x on L.H.S.is
$1 + 2 + 4 + . . . . + 128$

$= 1 + 2 + 2^{2} + . . . . + 2^{7}$

$= \frac{2^{8} - 1}{2 - 1}$

=255

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