Take a sphere and draw on it a great circle (a great circle is a circle whose centre is the centre of the sphere). There are two regions created. Here, I am referring to regions on the surface of the sphere. Now draw another great circle: there are four regions. Now draw a third, not passing through the points of intersection of the first two. How many regions? Here's the general question: How many regions are created by n great circles, no three concurrent, drawn on the surface of the sphere?

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Take a sphere and draw on it a great circle (a great circle is a circle whose centre is the centre of the sphere). There are two regions created. Here, I am referring to regions on the surface of the sphere. Now draw another great circle: there are four regions. Now draw a third, not passing through the points of intersection of the first two. How many regions? Here&#039;s the general question: How many regions are created by n great circles, no three concurrent, drawn on the surface of the sphere?

Tail3vn · Accepted Answer

Step 1Calculate the number of the regions for 3 non - collinear circles.For 1 great circle =2 regionsFor 2 great circles =4 regionsFor 3 great circles =8 regionsAs the number of the circles increase by 1, the number of regions increase by 2n times.Therefore, for (n+1)th circle the number of the regions increse by 2n times.The sequence is a quadratic sequence.Step 2Determine the quadratic sequence.Sequence for number of regions =2,4,8,14⋯Tn=an2+bn+cT1=2a(1)2+b(1)+c=2a+b+c=2 (1)T2=4a(2)2+b(2)+c=44a+2b+c=4 (2)T3=8a(3)2+b(3)+c=89a+3b+c=8 (3)Step 3After solving the 3 equations, the values for a=1, b=−1 and c=2.Tn=an2+bn+ca=1, b=−1 and c=2Tn=(1)n2+(−1)n+2Tn=n2−n+2Hence, the number of regions for the n circles isTn=n2−n+2where, n is the number of circleswhich are non collinear.

Take a sphere and draw on it a great circle (a great circle is a circle whose centre is the centre o

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