Given five numbers a, b, c, d and n (where a, b, c, d, n > 0). These values represent n terms of two series. The two series formed by these four numbers are b, b+a, b+2a….b+(n-1)a and d, d+c, d+2c, ….. d+(n-1)c

These two series will collide when at any single point summation values becomes exactly the same for both the series.Print the collision point.

Example: Input : a = 20, b = 2, c = 9, d = 19, n = 100 Output: 82 Explanation: Series1 = (2, 22, 42, 62, 82, 102...) Series2 = (28, 37, 46, 55, 64, 73, 82, 91..) So the first collision point is 82.

A **naive approach** is to calculate both the series in two different arrays, and then check for each element if it collides by running two nested loops

*Time complexity*: O(n * n)

*Auxiliary Space*: O(n)

** An efficient Approach **to the problem mentioned above is:

* Generate all elements of first series. Let current element be x.

* If x is also an element of second series, then following conditions should satisfy.

…..a) x should be greater than or equal to first element of second series.

…..a) Difference between x and first element should be divisible by c.

*If the above conditions are satisfied then the i-th value is the required meeting point.

Below is the implementation of the above problem :

## C++

`// CPP program to calculate the colliding ` `// point of two series ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` ` ` `void` `point(` `int` `a, ` `int` `b, ` `int` `c, ` `int` `d, ` `int` `n) ` `{ ` ` ` `int` `x , flag = 0; ` ` ` ` ` `// Iterating through n terms of the ` ` ` `// first series ` ` ` `for` `(` `int` `i = 0; i < n; i++) ` ` ` `{ ` ` ` ` ` `// x is i-th term of first series ` ` ` `x = b + i * a; ` ` ` ` ` `// d is first element of second ` ` ` `// series and c is common difference ` ` ` `// for second series. ` ` ` `if` `((x - d) % c == 0 and x - d >= 0) ` ` ` `{ ` ` ` `cout << x << endl ; ` ` ` `flag = 1; ` ` ` `break` `; ` ` ` `} ` ` ` ` ` `} ` ` ` ` ` `// If no term of first series is found ` ` ` `if` `(flag == 0) ` ` ` `{ ` ` ` `cout << ` `"No collision point"` `<< endl; ` ` ` `} ` ` ` ` ` `} ` ` ` ` ` `// Driver function ` `int` `main() ` `{ ` ` ` `int` `a = 20 ; ` ` ` `int` `b = 2 ; ` ` ` `int` `c = 9; ` ` ` `int` `d = 19; ` ` ` `int` `n = 20; ` ` ` `point(a, b, c, d, n); ` ` ` `return` `0; ` `} ` ` ` `// This code is contributed by 'saloni1297'. ` |

## Java

`// Java program to calculate the colliding ` `// point of two series ` `import` `java.io.*; ` ` ` `class` `GFG ` `{ ` ` ` `static` `void` `point(` `int` `a, ` `int` `b, ` `int` `c, ` `int` `d, ` `int` `n) ` ` ` `{ ` ` ` `int` `x , flag = ` `0` `; ` ` ` ` ` `// Iterating through n terms of the ` ` ` `// first series ` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++) ` ` ` `{ ` ` ` ` ` `// x is i-th term of first series ` ` ` `x = b + i * a; ` ` ` ` ` `// d is first element of second ` ` ` `// series and c is common difference ` ` ` `// for second series. ` ` ` `if` `((x - d) % c == ` `0` `&& x - d >= ` `0` `) ` ` ` `{ ` ` ` `System.out.println( x ) ; ` ` ` `flag = ` `1` `; ` ` ` `break` `; ` ` ` `} ` ` ` ` ` `} ` ` ` ` ` `// If no term of first series is found ` ` ` `if` `(flag == ` `0` `) ` ` ` `{ ` ` ` `System.out.println (` `"No collision point"` `); ` ` ` `} ` ` ` ` ` ` ` `} ` ` ` ` ` ` ` `// Driver function ` ` ` `public` `static` `void` `main (String[] args) ` ` ` `{ ` ` ` `int` `a = ` `20` `; ` ` ` `int` `b = ` `2` `; ` ` ` `int` `c = ` `9` `; ` ` ` `int` `d = ` `19` `; ` ` ` `int` `n = ` `20` `; ` ` ` `point(a, b, c, d, n); ` ` ` ` ` `} ` `} ` `// This code is contributed by vt_m ` |

## Python

`# Function to calculate the colliding point ` `# of two series ` `def` `point(a, b, c, d, n): ` ` ` ` ` `# Iterating through n terms of the ` ` ` `# first series ` ` ` `for` `i ` `in` `range` `(n): ` ` ` ` ` `# x is i-th term of first series ` ` ` `x ` `=` `b ` `+` `i` `*` `a ` ` ` ` ` `# d is first element of second ` ` ` `# series and c is common difference ` ` ` `# for second series. ` ` ` `if` `(x` `-` `d)` `%` `c ` `=` `=` `0` `and` `x` `-` `d >` `=` `0` `: ` ` ` `print` `x ` ` ` `return` ` ` ` ` `# If no term of first series is found ` ` ` `else` `: ` ` ` `print` `"No collision point"` ` ` ` ` `# Driver code ` `a ` `=` `20` `b ` `=` `2` `c ` `=` `9` `d ` `=` `19` `n ` `=` `20` `point(a, b, c, d, n) ` |

## C#

`// C# program to calculate the colliding ` `// point of two series ` `using` `System; ` ` ` `class` `GFG { ` ` ` ` ` `static` `void` `point(` `int` `a, ` `int` `b, ` `int` `c, ` ` ` `int` `d, ` `int` `n) ` ` ` `{ ` ` ` `int` `x, flag = 0; ` ` ` ` ` `// Iterating through n terms of the ` ` ` `// first series ` ` ` `for` `(` `int` `i = 0; i < n; i++) { ` ` ` ` ` `// x is i-th term of first series ` ` ` `x = b + i * a; ` ` ` ` ` `// d is first element of second ` ` ` `// series and c is common difference ` ` ` `// for second series. ` ` ` `if` `((x - d) % c == 0 && x - d >= 0) { ` ` ` `Console.WriteLine(x); ` ` ` `flag = 1; ` ` ` `break` `; ` ` ` `} ` ` ` `} ` ` ` ` ` `// If no term of first series is found ` ` ` `if` `(flag == 0) { ` ` ` `Console.WriteLine(` `"No collision point"` `); ` ` ` `} ` ` ` `} ` ` ` ` ` `// Driver function ` ` ` `public` `static` `void` `Main() ` ` ` `{ ` ` ` `int` `a = 20; ` ` ` `int` `b = 2; ` ` ` `int` `c = 9; ` ` ` `int` `d = 19; ` ` ` `int` `n = 20; ` ` ` `point(a, b, c, d, n); ` ` ` `} ` `} ` ` ` `// This code is contributed by vt_m ` |

## PHP

`<?php ` `// PHP program to calculate ` `// the colliding point of ` `// two series ` ` ` ` ` `function` `point(` `$a` `, ` `$b` `, ` `$c` `, ` `$d` `, ` `$n` `) ` `{ ` ` ` `$x` `; ` `$flag` `= 0; ` ` ` ` ` `// Iterating through ` ` ` `// n terms of the ` ` ` `// first series ` ` ` `for` `(` `$i` `= 0; ` `$i` `< ` `$n` `; ` `$i` `++) ` ` ` `{ ` ` ` ` ` `// x is i-th term ` ` ` `// of first series ` ` ` `$x` `= ` `$b` `+ ` `$i` `* ` `$a` `; ` ` ` ` ` `// d is first element of ` ` ` `// second series and c is ` ` ` `// common difference for ` ` ` `// second series. ` ` ` `if` `((` `$x` `- ` `$d` `) % ` `$c` `== 0 ` `and` ` ` `$x` `- ` `$d` `>= 0) ` ` ` `{ ` ` ` `echo` `$x` `; ` ` ` `$flag` `= 1; ` ` ` `break` `; ` ` ` `} ` ` ` ` ` `} ` ` ` ` ` `// If no term of first ` ` ` `// series is found ` ` ` `if` `(` `$flag` `== 0) ` ` ` `{ ` ` ` `echo` `"No collision po$"` `; ` ` ` `} ` ` ` ` ` `} ` ` ` ` ` `// Driver Code ` ` ` `$a` `= 20 ; ` ` ` `$b` `= 2 ; ` ` ` `$c` `= 9; ` ` ` `$d` `= 19; ` ` ` `$n` `= 20; ` ` ` `point(` `$a` `, ` `$b` `, ` `$c` `, ` `$d` `, ` `$n` `); ` ` ` `// This code is contributed by anuj_67. ` `?> ` |

**Output :**

82

Time complexity : O(n)

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