Define $f_{n}(x) = (1 + x)(1 + 2x)(1 + 4x)\ldots\left( 1 + 2^{n}x \right) = a_{n,0} + a_{n,1}\ ^{x}

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Define $f_{n}(x) = (1 + x)(1 + 2x)(1 + 4x)\ldots\left( 1 + 2^{n}x \right) = a_{n,0} + a_{n,1}\ ^{x} + a_{n,2}x^{2} + \ldots + a_{n,n}x^{n}$, where n is a positive integer, then

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Sol. [Key Concept: Simplify the defined sum $S_{n}$.]$$a_{n},2 = \sum_{i = 0}^{n - 1}\mspace{2mu}\sum_{j = i + 1}^{n}\mspace{2mu} 2^{i}2^{j} = \frac{1}{2}\left\lbrack \left( 1 + 2 + 2^{n} + \ldots + 2^{n} \right)^{2} - \left( 1^{2} + 2^{2} + 2^{4} + \ldots + 2^{2n} \right) \right\rbrack$$$$= \frac{\left( 2^{n + 1} - 1 \right)\left( 2^{n + 1} - 2 \right)}{3}$$✘ (a) not the derived value✔ (b) $a_{100,2} = \frac{{(2}^{101}{- 1)(2}^{101} - 2)}{3}$✘ (c) not the derived value✔ (d) $a_{100,2} - a_{99,2}{= 2}^{200}{- 2}^{100}$

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