# Sum of series M/1 + (M+P)/2 + (M+2*P)/4 + (M+3*P)/8……up to infinite

Find the sum of series M/1 + (M+P)/2 + (M+2*P)/4 + (M+3*P)/8……up to infinite where M and P are positive integers.

Examples:

Input : M = 0, P = 3; Output : 6 Input : M = 2, P = 9; Output : 22

** Method : **

**S = M/1 + (M + P)/2 + (M + 2*P)/4 + (M + 3*P) / 8……up to infinite**

so the solution of this series will be like this

we are going to divide this series into two parts-

S = (M/1 + M/2 + M/4 + M/8……up to infinite) + ( p/2 + (2*p)/4 + (3*p)/8 + ….up to infinite)

let us consider it

**S = A + B ……..eq(1)**

where,

A = M/1 + M/2 + M/4 + M/8……up to infinite

A = M*(1 + 1/2 + 1/4 + 1/8….up to infinite)

which is G.P of infinite terms with r = 1/2;

According to the formula of G.P sum of infinite terms for r < 1 and

a is first term and r is common ratio so now,

A = M * ( 1 / (1 – 1/2) )

A = 2 * M ;

**Now for B –**

B = ( p/2 + (2*p)/4 + (3*p)/8 + ….up to infinite)

B = P/2 * ( 1 + 2*(1/2) + 3*(1/4) + ……up to infinite)

it is sum of AGP of infinite terms with a = 1, r = 1/2 and d = 1;

According to the formula where a is first term,

r is common ratio and d is common difference so now,

B = P/2 * ( 1 / (1-1/2) + (1*1/2) / (1-1/2)^2 )

B = P/2 * 4

B = 2*P ;

put value of A and B in eq(1)

**S = 2(M + P)**

## C++

`#include <iostream> ` `using` `namespace` `std; ` ` ` `int` `sum(` `int` `M, ` `int` `P) ` `{ ` ` ` `return` `2*(M + P); ` `} ` ` ` `// driver code ` `int` `main() { ` ` ` ` ` `int` `M = 2, P = 9; ` ` ` `cout << sum(M,P); ` ` ` `return` `0; ` `} ` |

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## Java

`// javaProgram to finding the ` `// sum of the series ` `import` `java.io.*; ` ` ` `class` `GFG { ` ` ` ` ` `// function that calculate ` ` ` `// the sum of the nth series ` ` ` `static` `int` `sum_series(` `int` `M, ` `int` `P) ` ` ` `{ ` ` ` `return` `2` `* (M + P); ` ` ` `} ` ` ` ` ` `// Driver function ` ` ` `public` `static` `void` `main (String[] args) ` ` ` `{ ` ` ` `int` `M = ` `2` `; ` ` ` `int` `P = ` `9` `; ` ` ` `System.out.println( sum_series(M, P)) ; ` ` ` `} ` `} ` |

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## Python

`# Python 3 Program to finding ` `# the sum of the series ` ` ` `# function that calculate ` `# the sum of the series ` `def` `sum_series(M, P): ` ` ` ` ` `return` `int` `(` `2` `*` `(M ` `+` `P)) ` ` ` `# Driver function ` `M ` `=` `2` `P ` `=` `9` `print` `(sum_series(M ,P)) ` |

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## C#

`// C# program to finding the ` `// sum of the series ` `using` `System; ` ` ` `class` `GFG { ` ` ` ` ` `// Function that calculate ` ` ` `// the sum of the nth series ` ` ` `static` `int` `sum_series(` `int` `M, ` `int` `P) ` ` ` `{ ` ` ` `return` `2*(M + P); ` ` ` `} ` ` ` ` ` `// Driver Code ` ` ` `public` `static` `void` `Main () ` ` ` `{ ` ` ` `int` `M =2; ` ` ` `int` `P =9; ` ` ` ` ` `Console.Write( sum_series(M,P)) ; ` ` ` `} ` `} ` |

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## PHP

`<?php ` `// PHP program to finding the ` `// sum of the series ` ` ` `// Function that calculate ` `// the sum of the nth series ` `function` `sum(` `$M` `, ` `$P` `) ` `{ ` ` ` `return` `2*(` `$M` `+ ` `$P` `); ` `} ` ` ` `// Driver Code ` `$M` `= 2; ` `$P` `= 9; ` `echo` `sum(` `$M` `, ` `$P` `); ` ` ` `// This code is contributed by mits ` `?> ` |

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**Output:**

22

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