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 November 2nd, 2013, 12:56 PM #1 Newbie   Joined: Nov 2013 Posts: 1 Thanks: 0 Total area of circle using sequences and or series How do I find the total area of total area of the red circles C_n,n=1,2,3? I am very confused with this problem. I have been working on it for about an hour and still nothing . Help would be greatly appreciated! I know that I can use sequences to solve this problem, however, I am confused on how to approach it. I pasted a url that displays the picture of my question. http://tinypic.com/r/2dkmmq1/5
 November 2nd, 2013, 03:25 PM #2 Senior Member   Joined: Jul 2011 Posts: 118 Thanks: 0 Re: Total area of circle using sequences and or series For the first circle we have: $(R+r_1)^2-R^2=(R-r_1)^2\Rightarrow r_1=\frac{R}{4}$ For the next ones the formula is: $\sqrt{(R+r_{n+1})^2-R^2}+r_{n+1}=R-2\sum_{k=1}^nr_k$ First few results are: $r_2=\frac{R}{12} r_3=\frac{R}{24} r_4=\frac{R}{40}$ So we can guess that: $r_n=\frac{R}{2n(n+1)}$ It is easy to show that by induction: $r_1=\frac{R}{4} R-2\sum_{k=1}^nr_k=R-2\sum_{k=1}^n$$\frac{R}{2n}-\frac{R}{2(n+1)}$$=\frac{R}{n+1} \sqrt{(R+r_{n+1})^2-R^2}+r_{n+1}=\frac{R}{n+1} (R+r_{n+1})^2-R^2=$$\frac{R}{n+1}-r_{n+1}$$^2 r_{n+1}=\frac{R}{2(n+1)(n+2)}$

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