# Thread: Infinite summations: area of polygon

1. ## Infinite summations: area of polygon

The vertices of a polygon are:

$(-1,0), \ (-1+2^{-n},1-(-1+2^{-n})^{2}), \ (-1+2 \cdot 2^{-n},1-(-1+2\cdot2^{-n})^{2}), \ (-1+3\cdot2^{-n},1-(-1+3\cdot2^{-n})^{2}), ...,(1,0)$

I have to show that the area of the polygon is 1 + 4^(-1) + 4^(-2) + ... + 4^(-n).

How would I go about doing that? Not sure how to proceed.

2. ## Re: Infinite summations: area of polygon

Originally Posted by BrownianMan
The vertices of a polygon are:

$(-1,0), \ (-1+2^{-n},1-(-1+2^{-n})^{2}), \ (-1+2 \cdot 2^{-n},1-(-1+2\cdot2^{-n})^{2}), \ (-1+3\cdot2^{-n},1-(-1+3\cdot2^{-n})^{2}), ...,(1,0)$

I have to show that the area of the polygon is 1 + 4^(-1) + 4^(-2) + ... + 4^(-n).

How would I go about doing that? Not sure how to proceed.
Use the Polygon - Wikipedia, the free encyclopedia formula.

3. ## Re: Infinite summations: area of polygon

I tried using it and could not get the answer.

4. ## Re: Infinite summations: area of polygon

Originally Posted by BrownianMan
The vertices of a polygon are:

$(-1,0), \ (-1+2^{-n},1-(-1+2^{-n})^{2}), \ (-1+2 \cdot 2^{-n},1-(-1+2\cdot2^{-n})^{2}), \ (-1+3\cdot2^{-n},1-(-1+3\cdot2^{-n})^{2}), ...,(1,0)$

I have to show that the area of the polygon is 1 + 4^(-1) + 4^(-2) + ... + 4^(-n)
I don't know how in the world anyone can read the above.

5. ## Re: Infinite summations: area of polygon

What's wrong with it?

6. ## Re: Infinite summations: area of polygon

Does anyone have any idea on how to solve this? I tried using the formula for the general polygon, but I can't get 1 + 4^(-1) + 4^(-2) + ... + 4^(-n) for the area.