If `lim_(x rarr 1)((x+x^2+x^3+....+x^n-n)/(x-1))=820`, then find n.

Options

A. n
B. `(n+1)/(2)`
C. `(n(n+1))/(2)`
D. `(n(n-1))/(2)`

Answer

C

Explanation

We have
`lim_(xto1)(x+x^2+.....+x^n-n)/(x-1)`
`=lim_(x to 1) ((x-1)+(x^2-1^2)+(x^3-1^3)+.......+(x^n-1^n))/(x-1)`
`lim_(x to 1) {(x-1)/(x-1)+(x^2-1^2)/(x-1)+(x^3-1^3)/(x-1)+.....(x^n-1^n)/(x-1)}`
`=1+2(1)^2-1+3(1)^3-1+....+n(1)^n-1`
` =1+2+3+.......+n=(n+1))/(2)`