- #1

utkarshakash

Gold Member

- 855

- 13

## Homework Statement

Let An = [itex]\sum_{r=1}^n arccos \frac{r}{n} [/itex] and Bn=[itex]\sum_{r=0}^{n-1} arccos \frac{r}{n} [/itex] for n=1,2,3....... then

A)A(2010) < 2010 and B(2010)>2010

B)A(2010) > 2010 and B(2010)<2010

C)A(n) < B(n) for all n

D)[itex]\lim_{n \to \infty} \dfrac{A_n}{n} = 1 [/itex]

More than one option is correct.

## The Attempt at a Solution

Finding A(2010) seems really challenging. I start by writing out all the terms manually.

arccos(1/2010)+arccos(2/2010)+arccos(3/2010)+arccos(4/2010)+..... ..arccos(2010/2010)

Atmost I can only simplify the last term to 0. But what about the rest of the terms? The formula arccosx + arccosy = arccos{xy − √[(1 − x2)(1 − y2)]} doesn't help. It only makes the terms more complicated.