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Expansions to Remove Indeterminate Form

If 1+x1+x21...

Question

If $(1 + x) (1 + x^{2}) (1 + x^{4}) . . . . (1 + x^{128}) = \sum_{r = 0}^{n} x^{r}$ then findthe value of $n$ .

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Solution

$(1 + x) (1 + x^{2}) (1 + x^{4}) . . . . . (1 + x^{128}) = n \sum r = 0 x^{r}$
$(1 + x^{2 n})$ so $n = \frac{128}{2} = 64$

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