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If $\lim\limits_{x\to1}{x+x^2+x^3+.....+x^n-n\over x-1}=820, (n\in N)$ then the value of n is equal to .......

Numerical Value Type

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Sol. $\lim\limits_{x\to1}{x+x^2+x^3+.....+x^n-n\over x-1}=820\left({0\over0}\right)$

$\lim\limits_{x\to1}{x+2x+3x^2+.....+nx^{n-1}\over1}=820$

$\implies1+2+3+.....+n=820$

$\implies{n(n+1)\over2}=820$

$\implies n^2+n=1640$

$\implies n^2+n-1640=0$

$\implies n=40\quad\quad n\in N$

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