If lim x→1 x + x^2 + x^3 +.....+ x^n − n / x − 1 = 820, (n ∈ N) then the value of n is equal to .......

Question

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

Answer 1

Ans. (40.00)

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

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

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

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

$\implies n^2+n=1640$

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

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