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Factorisation by Regrouping Terms

If 1+x1+x21...

Question

If $(1 + x) (1 + x^{2}) (1 + x^{4}) . . . .$ $(1 + x^{128}) = (1 + x + x^{2} . . . . . x^{n})$ then $n = ?$
Given $| x | < 1$

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Solution

$\to$ Highest power of x
would be $1 + 2 + 4 + - - - + 128$
$= 2^{0} + 2^{1} + 2^{2} + - - - + 2^{7}$
$\Rightarrow$ Sum of G.P. with $a = 2^{\circ}$ , r = 2, n = 8
$\Rightarrow S = \frac{1 (2^{8} - 1)}{2 - 1} = 256 - 1 = 255$

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