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Respected Sir, Please give a complete information about the following one:

Find a recurrence relation for the number of regions created by 'n' mutually intersecting circles on a piece of paper (no three circles have a common intersecting point).

if you could be a clear a possible with your answer that would be awesome... thanks.

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And respected Madams? Anyway, do you mean the maximum number of regions? I'm not sure if that can be put into a simple recurrence. – anon Jul 27 '11 at 9:31

Google search for recurrence regions circles led me to this document.

Example 1.1 gives a derivation of formulas $a_n=a_{n-1}+2(n-1)$ and $a_n=n^2-n+2$. (If I understood your question correctly, it is identical to this example.)

BTW the google search provided also this answer, which seems to be incorrect.

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I found the answer to be a sub n = a sub n minus 1 + n. it works for the first 5 initial conditions (by drawing out the circles and by inspection). i agree that the google answer is wrong, because then a sub 3 could not equal 6 (even though it does) because 2 cubed is 8, not 6.

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what is a sub n? please try to be a little bit more clear – hHhh Jun 26 '15 at 15:00
I am not sure how to explain what a sub n is. a sub n is what were trying to solve for, in this case, a sub n = the number of regions created by n mutually intersecting circles on a piece of paper. – mathematician Jun 27 '15 at 14:48

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