The coefficient of x^n in the product (1-x)(1-2 x)(1-2^2 ·x)(1-2^3 ·x) …(1-2^n+x) is equal to (

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The coefficient of x^n in the product (1-x)(1-2 x)(1-2^2...

The coefficient of $x^{n}$ in the product $(1-x)(1-2 x)\left(1-2^{2} \cdot x\right)\left(1-2^{3} \cdot x\right) \ldots\left(1-2^{n}+x\right)$ is equal to $($ A $)\left(1-2^{n+1}\right) 2^{\frac{n(n-1)}{2}}$. (B) $\left(2^{n+1}-1\right) \cdot 2^{\frac{n(n-1)}{2}}$ (C) $\left(1-2^{n}\right) 2^{\frac{n(n-1)}{2}}$. (D) None of these

The coefficient of $x^{n}$ in the product $(1-x)(1-2 x)\left(1-2^{2} \cdot x\right)\left(1-2^{3} \cdot x\right) \ldots\left(1-2^{n}+x\right)$ is
equal to
$($ A $)\left(1-2^{n+1}\right) 2^{\frac{n(n-1)}{2}}$.
(B) $\left(2^{n+1}-1\right) \cdot 2^{\frac{n(n-1)}{2}}$
(C) $\left(1-2^{n}\right) 2^{\frac{n(n-1)}{2}}$.
(D) None of these

Key Concepts

Polynomial Expansion

This concept involves writing a product of linear factors as a sum of terms, each involving a power of the variable. It requires multiplying out expressions like (1 - ax) systematically, and keeping track of the contributions each term makes to the desired power of the variable.

Coefficient Extraction

Coefficient extraction is the process of determining the numerical multiplier of a specific power of the variable in a fully expanded polynomial. In problems involving products of linear factors, it typically involves summing over all possible selections of terms from each factor that multiply together to yield the target exponent.

Combinatorial Selection

When expanding a product of several factors, obtaining a specific power of the variable often requires choosing one term from each factor in specific combinations. The combinatorial approach involves identifying the different ways of selecting these terms (for instance, by choosing which factors contribute the variable term and which contribute the constant term) and then summing their contributions.

Geometric Series

A geometric series arises naturally in problems where terms involve constant ratios, such as powers of a constant multiplying the variable. Recognizing and summing a geometric series can simplify the expression for the coefficient when it involves sums over exponentials with a common base.

Exponential Arithmetic

The manipulation and combination of exponents is key in problems where the coefficients include powers of a number. Understanding rules of exponents (such as the product rule and sum of arithmetic progressions in the exponents) is essential for simplifying and expressing the final answer in compact form.

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