Geometric sequence and calculus (1)

Algebra Level 3

Let \(x+x^2+x^3+\cdots+x^{n-2}+x^{n-1}+x^{n}=\dfrac{x(x^n-1)}{x-1}\). (\(x\neq1\))

What is \(1+2x+3x^2+ \cdots + (n-2)x^{n-3}+(n-1)x^{n-2}+nx^{n-1}\)?


A. \(\quad \dfrac{nx^{n+1}-nx^n-x^n+1}{x^2-2x+1}\)

B. \(\quad \dfrac{x^{n+2}-x^{n+1} }{x^2-1}\)

C.\(\quad \dfrac{(n+1)x^{n+1}-n^2x^n+x^n+1}{x^2-2x+1}\)

D.\(\quad \dfrac{nx^{n+1}-nx^n-x^n-1}{x^2-3x+2}\)

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