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Sum of n Terms

If 1+x1+x21+x...

Question

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

A

256

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B

255

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C

254

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D

257

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Solution

The correct option is B $255$
$(1 + x) (1 + x^{2}) (1 + x^{4}) \dots (1 + x^{128}) = n \sum r = 0 x^{r}$
Highest degree of $x$ on both sides,
$n = 1 + 2 + 4 + \dots \dots + 128 \Rightarrow n = 1 + 2 + 2^{2} + \dots \dots + 2^{7} \Rightarrow n = \frac{2^{8} - 1}{2 - 1} ∴ n = 255$

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