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 \frac{a}{1-r} 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 \frac{a}{1-r}+\frac{dr}{(1-r)^{2}} 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++

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#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

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// 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

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# 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#

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// 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

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<?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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Improved By : Mithun Kumar