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

Step 1
Calculate the number of the regions for 3 non - collinear circles.
For 1 great circle ${\displaystyle =2}$ regions
For 2 great circles ${\displaystyle =4}$ regions
For 3 great circles ${\displaystyle =8}$ regions
As the number of the circles increase by 1, the number of regions increase by 2n times.
Therefore, for ${\displaystyle {(n+1)}^{th}}$ 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 ${\displaystyle =2,4,8,14\cdots }$
${\displaystyle T_n=a{n}^2+bn+c}$
${\displaystyle T_1=2}$
${\displaystyle a{\left(1\right)}^2+b\left(1\right)+c=2}$
${\displaystyle a+b+c=2}$ (1)
${\displaystyle T_2=4}$
${\displaystyle a{\left(2\right)}^2+b\left(2\right)+c=4}$
${\displaystyle 4a+2b+c=4}$ (2)
${\displaystyle T_3=8}$
${\displaystyle a{\left(3\right)}^2+b\left(3\right)+c=8}$
${\displaystyle 9a+3b+c=8}$ (3)
Step 3
After solving the 3 equations, the values for ${\displaystyle a=1,b=-1}$ and ${\displaystyle c=2}$.
${\displaystyle T_n=a{n}^2+bn+c}$
${\displaystyle a=1,b=-1}$ and ${\displaystyle c=2}$
${\displaystyle T_n=\left(1\right)n^2+(-1)n+2}$
${\displaystyle T_n=n^2-n+2}$
Hence, the number of regions for the n circles is
$T_n=n^2-n+2$
where, n is the number of circleswhich are non collinear.