# Maximum number of region in which N non-parallel lines can divide a plane

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.

**Examples:**

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 n^{th} 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:

## C++

`// C++ program to implement the above problem ` ` ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find the maximum ` `// number of regions on a plane ` `void` `maxRegions(` `int` `n) ` `{ ` ` ` `int` `num; ` ` ` `num = n * (n + 1) / 2 + 1; ` ` ` ` ` `// print the maximum number of regions ` ` ` `cout << num; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `n = 10; ` ` ` ` ` `maxRegions(n); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to implement the above problem ` `class` `GFG ` `{ ` ` ` ` ` `// Function to find the maximum ` ` ` `// number of regions on a plane ` ` ` `static` `void` `maxRegions(` `int` `n) ` ` ` `{ ` ` ` `int` `num; ` ` ` `num = n * (n + ` `1` `) / ` `2` `+ ` `1` `; ` ` ` ` ` `// print the maximum number of regions ` ` ` `System.out.println(num);; ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{ ` ` ` `int` `n = ` `10` `; ` ` ` `maxRegions(n); ` ` ` `} ` `} ` ` ` `// This code is contributed by 29AjayKumar ` |

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## Python3

`# Python3 program to implement ` `# the above problem ` ` ` `# Function to find the maximum ` `# number of regions on a plane ` `def` `maxRegions(n): ` ` ` `num ` `=` `n ` `*` `(n ` `+` `1` `) ` `/` `/` `2` `+` `1` ` ` ` ` `# print the maximum number ` ` ` `# of regions ` ` ` `print` `(num) ` ` ` `# Driver code ` `n ` `=` `10` ` ` `maxRegions(n) ` ` ` `# This code is contributed ` `# by Mohit Kumar ` |

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## C#

`// C# program to implement the above problem ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` ` ` `// Function to find the maximum ` ` ` `// number of regions on a plane ` ` ` `static` `void` `maxRegions(` `int` `n) ` ` ` `{ ` ` ` `int` `num; ` ` ` `num = n * (n + 1) / 2 + 1; ` ` ` ` ` `// print the maximum number of regions ` ` ` `Console.WriteLine(num); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main(String[] args) ` ` ` `{ ` ` ` `int` `n = 10; ` ` ` `maxRegions(n); ` ` ` `} ` `} ` ` ` `// This code is contributed by 29AjayKumar ` |

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**Output:**

56

** Time Complexity : ** O(1)

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