Given N‘, the number of non-parallel lines. The task is to find the maximum number of regions in which these lines can divide a plane.
Input : N = 3
Output : 7
Input : N = 2
Output : 4
Approach : The above image shows the maximum number of regions a line can divide a plane. One line can divide a plane into two regions, two non-parallel lines can divide a plane into 4 regions and three non-parallel lines can divide into 7 regions and so on. When the nth line is added to a cluster of (n-1) lines then the maximum number of extra regions formed is equal to n.
Now solve the recursion as follows:
L(2) – L(1) = 2 … (i)
L(3) – L(2) = 3 … (ii)
L(4) – L(3) = 4 … (iii)
. . .
. . .
L(n) – L(n-1) = n ; … (n)
Adding all the above equation we get,
L(n) – L(1) = 2 + 3 + 4 + 5 + 6 + 7 + …… + n ;
L(n) = L(1) + 2 + 3 + 4 + 5 + 6 + 7 + …… + n ;
L(n) = 2 + 2 + 3 + 4 + 5 + 6 + 7 + …… + n ;
L(n) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + …… + n + 1 ;
L(n) = n ( n + 1 ) / 2 + 1 ;
The number of region in which N non-parallel lines can divide a plane is equal to N*( N + 1 )/2 + 1.
Below is the implementation of the above approach:
Time Complexity : O(1)
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