That's a Parabola!

Geometry Level 5

Consider a Parabola

y=x24y = \frac{x^{2}}{4}

With vertex 'O'

And a point F(0 , 1)

Let B1(x1,y1),B2(x2,y2)Bn(xn,yn)B_1( x_1 , y_1 ) , B_2( x_2 , y_2 ) \ldots B_n( x_n , y_n )

are 'n' points on the Parabola

Such that xk>0x_k > 0 and

OFBk=kπ2n \displaystyle \angle OFB_k = \frac{ k \pi}{2n} ( k = 1, 2, 3 \ldots , n)

Then find the value of

limn1nk=1nFBk\lim_{ n \to \infty } \frac{1}{n} \sum_{k = 1}^{n} FB_k

upto 3 decimal place

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