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

Phoebe Xiong 2022-03-03 Answered
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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Expert Answer

Tail3vn
Answered 2022-03-04 Author has 3 answers

Step 1
Calculate the number of the regions for 3 non - collinear circles.
For 1 great circle =2 regions
For 2 great circles =4 regions
For 3 great circles =8 regions
As 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 2
Determine the quadratic sequence.
Sequence for number of regions =2,4,8,14
Tn=an2+bn+c
T1=2
a(1)2+b(1)+c=2
a+b+c=2 (1)
T2=4
a(2)2+b(2)+c=4
4a+2b+c=4 (2)
T3=8
a(3)2+b(3)+c=8
9a+3b+c=8 (3)
Step 3
After solving the 3 equations, the values for a=1, b=1 and c=2.
Tn=an2+bn+c
a=1, b=1 and c=2
Tn=(1)n2+(1)n+2
Tn=n2n+2
Hence, the number of regions for the n circles is
Tn=n2n+2
where, n is the number of circleswhich are non collinear.

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