Geometric sequence and calculus(2)

Calculus Level pending

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 \(2+6x+12x^2+\cdots+\) \((n^2-5n+6)x^{n-4}+(n^2-3n+2)x^{n-3}+(n^2-n)x^{n-2} ?\)


A.\(\frac{(n^2-n)x^{n+2}-(2n^2-2)x^{n+1}+(n^2+n)x^{n}-2}{(x-1)^4}\)

B.\(\frac{(n^2+n)x^{n+1}-(2n^2+2)x^n+(n^2-n)x^{n-1}+2}{(x-1)^3}\)

C.\(\frac{(n^2-n)x^{n+1}-(2n^2-2)x^n+(n^2+n)x^{n-1}-2}{(x-1)^3}\)

(Calculate and choose the answer carefully)

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