Geometric sequence and calculus (1)

Algebra Level 3

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

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

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

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

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

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

A C D All of them are incorrect B

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Tommy Li by Tommy Li , 22, Hong Kong

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1 solution

Tommy Li
May 25, 2016

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

Differentiate both sides w . r . t . x w.r.t. x

1 + 2 x + 3 x 2 + . . . + ( n − 2 ) x n − 3 + ( n − 1 ) x n − 2 + n x n − 1 1+2x+3x^2+...+(n-2)x^{n-3}+(n-1)x^{n-2}+nx^{n-1} = ( x − 1 ) ( ( n + 1 ) x n − 1 ) − ( x n + 1 − x ) ( 1 ) ( x − 1 ) 2 \frac{(x-1)((n+1)x^n-1)-(x^{n+1} -x)(1)}{(x-1)^{2}}

= n x n + 1 − n x n − x n + 1 x 2 − 2 x + 1 \frac{nx^{n+1}-nx^n-x^n+1}{x^2-2x+1}

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